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Introduction {#intro .unnumbered}
============

In this work I will construct certain general bundles $\langle\mathfrak{M},\rho,X\rangle$ and $\langle\mathfrak{B},\eta,X\rangle$ of Hausdorff locally convex spaces associated to a given Banach bundle $\langle\mathfrak{E},\pi,X\rangle$. Then I will present conditions ensuring the existence of bounded selections $\mathcal{U}\in
\Gamma^{x_{\infty}}(\rho)$ and $\mathcal{P}\in
\Gamma^{x_{\infty}}(\eta)$ both continuous at a point $x_{\infty}\in X$, such that $\mathcal{U}(x)$ is a $C_{0}-$semigroup of contractions on $\mathfrak{E}_{x}$ and $\mathcal{P}(x)$ is a spectral projector of the infinitesimal generator of the semigroup $\mathcal{U}(x)$, for every $x\in X$. In a subsequent paper I shall produce examples of the general results presented here.

Here ${\mathfrak{W}}
\doteqdot
{\left\langle {\mathfrak{M}},\rho,X{\right\rangle}}$ and ${\left\langle {\mathfrak{B}},\eta,X{\right\rangle}}$ are special kind of bundles of Hausdorff locally convex spaces (bundle of $\Omega-$spaces) while ${\mathfrak{V}}
\doteqdot
{\left\langle {\mathfrak{E}},\pi,X{\right\rangle}}$ is a suitable Banach bundle such that the common base space $X$ is a metrizable space. Moreover for all $x\in X$ the stalk ${\mathfrak{M}}_{x}
\doteqdot
\overset{-1}{\rho}(x)$ is a topological subspace of the space ${\mathcal{C}_{c}
\left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)}$ with the topology of compact convergence, of all continuous maps defined on ${\mathbb {R}}^{+}$ and with values in ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$, and the stalk ${\mathfrak{B}}_{x}
\doteqdot
\overset{-1}{\eta}(x)$ is a topological subspace of ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$. Here ${\mathfrak{E}}_{x}
\doteqdot
\overset{-1}{\pi}(x)$, while ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$, is the space, of all linear bounded maps on ${\mathfrak{E}}_{x}$ with the topology of uniform convergence over the subsets of $S_{x}\subset 
Bounded({\mathfrak{E}}_{x})$ which depends, for all $x\in X$, on the same subspace ${\mathcal{E}}\subseteq\Gamma(\pi)$. Finally $\rho:{\mathfrak{M}}\to X$, $\eta:{\mathfrak{B}}\to X$, and $\pi:{\mathfrak{E}}\to X$ are the projection maps of the respecive bundles and $\Gamma^{x_{\infty}}(\rho)$ is the class of all bounded selections, i.e. maps belongings to the set $\prod_{x\in X}{\mathfrak{M}}_{x}$ continuous at $x_{\infty}$ with respect to the topology on the bundle space ${\mathfrak{M}}$, similarly for $\Gamma^{x_{\infty}}(\eta)$.

**A fundamental remark is that the continuity at $x_{\infty}$ of ${\mathcal{U}}$ and ${\mathcal{P}}$ derives by a sort of continuity at the same point of the selection ${\mathcal{T}}$ of the graphs of the infinitesimal generators of the semigroups ${\mathcal{U}}$**, where this sort of continuity has to be understood in the following sense. For every $x\in X$ let ${\mathcal{T}}(x)$ be the graph of the infinitesimal generator $T_{x}$ of the semigroup ${\mathcal{U}}(x)$, then $$\label{19240703}
\begin{cases}
{\mathcal{T}}(x_{\infty})
=
\left\{
\phi(x_{\infty})
\mid
\phi
\in
\Phi
\right\}
\\
\Phi
\subseteq
\Gamma^{x_{\infty}}(\pi_{{\mathbf{E}}^{\oplus}})
\\
(\forall x\in X)
(\forall \phi\in\Phi)
(\phi(x)\in{\mathcal{T}}(x)),
\end{cases}$$ where $\Gamma^{x_{\infty}}(\pi_{{\mathbf{E}}^{\oplus}})$ is the class of all bounded selections of the direct sum of bundles ${\mathfrak{V}}\oplus{\mathfrak{V}}$ which are continuous at $x_{\infty}$.

Hence for any $v\in Dom(T_{x_{\infty}})$ there exits a bounded selection $\phi$ of ${\mathfrak{V}}\oplus{\mathfrak{V}}$ such that $$\label{16490703}
\begin{cases}
(v,T_{x_{\infty}}v)
=
\lim_{x\to x_{\infty}}
(\phi_{1}(x),\phi_{2}(x))
\\
(\phi_{1}(x),\phi_{2}(x))
\in Graph(T_{x}),
\forall x\in X-\{x_{\infty}\},
\end{cases}$$ where the limit is with respect to the topology on the bundle space of ${\mathfrak{V}}\oplus{\mathfrak{V}}$ [^1].

 $\left(\Theta,{\mathcal{E}}\right)-$structure.\
Relation between the topologies on ${\mathfrak{M}}$ and ${\mathfrak{B}}$ and that on ${\mathfrak{E}}$.  {#leftthetamathcaleright-structure.-relation-between-the-topologies-on-mathfrakm-and-mathfrakb-and-that-on-mathfrake. .unnumbered}
-------------------------------------------------------------------------------------------------------

**The main general strategy for obtaining the continuity at $x_{\infty}$ of ${\mathcal{U}}$ and ${\mathcal{P}}$, it is to correlate the topologies on the bundles spaces involved, among others those on ${\mathfrak{M}}$ and ${\mathfrak{B}}$, with that on the space ${\mathfrak{E}}$.** Due to this fact it is clear that in this work the construction of the right structures has a prominent role.

It is a well-known fact the relative freedom of choice of the topology on the bundle space of any bundle of $\Omega-$spaces. More exactly the possibility of choosing a linear subspace, which is the entire space if $X$ is compact, of the space of all (global) sections of the bundle, i.e. the space of all everywhere defined bounded continuous selections, see [@gie Theorem $5.9$]. This freedom of choice allows the construction of examples of some of the cited correlations of topologies.

Without entering in the definition of the topology of a bundle of $\Omega-$space, we can appreciate how much important it is to choose the “right” set of all sections (in symbols $\Gamma(\zeta)$) of a general bundle ${\left\langle {\mathfrak{Q}},\zeta,X{\right\rangle}}$ of $\Omega-$space, by the following simple but fundamental result, Corollary \[28111707\]. Let $f\in\prod_{x\in X}^{b}{\mathfrak{Q}}_{x}$ be any bounded selection and $x_{\infty}\in X$ such that there exists a section $\sigma\in\Gamma(\zeta)$ such that $\sigma(x_{\infty})=f(x_{\infty})$. Then by setting $f\in\Gamma^{x_{\infty}}(\zeta)$ iff $f$ is bounded and continuous at $x_{\infty}$ we have $$\label{21040503}
f\in\Gamma^{x_{\infty}}(\zeta)
\Leftrightarrow
(\forall j\in J)
(\lim_{z\to x_{\infty}}
\nu_{j}^{z}(f(z)-\sigma(z))=0),$$ where $J$ is a set such that $\{\nu_{j}^{z}\mid j\in J\}$ is a fundamental set of seminorms of the locally convex space ${\mathfrak{Q}}_{z}\doteqdot\overset{-1}{\zeta}(z)$ for all $z\in X$. About the problem of establishing if there are sections intersecting $f$ in $x_{\infty}$, we can use an important result of the theory of Banach bundles, stating that any Banach bundle over a locally compact base space is “full”, i.e. for any point of the bundle space there exists a section passing on it. For more general bundle of $\Omega-$space we can use the freedom before mentioned.

*The criterium I used for determining the correlations between ${\mathfrak{M}}$ (resp.${\mathfrak{B}}$) and ${\mathfrak{E}}$ is that of extending to a general bundle two properties of the topology of the space ${\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{s}(Z)\right)}$*.

Here $Z$ is a normed space, $S$ is a class of bounded subsets of $Z$, ${\mathcal{L}}_{s}(Z)$ is the space of all linear continuous maps on $Z$ with the pointwise topology, finally ${\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{s}(Z)\right)}$ is the space of all continuous maps on $Y$ with values in ${\mathcal{L}}_{s}(Z)$ with the topology of uniform convergence over the compact subsets of $Y$.

In order to simplify the notations we here shall consider $Z$ as a Banach space thus ${\mathcal{L}}_{S}(Z)=B_{s}(Z)$, i.e. the space of all bounded linear operators on $Z$ with the strong operator topology.

Let $X$ be a compact space, $Y$ a topological space $$\begin{aligned}
{\mathcal{M}}
\doteqdot
&
\{
F\in{\mathcal{C}_{b}
\left(X,{\mathcal{C}_{c}
\left(Y,B_{s}(Z)\right)}\right)}
\mid
(\forall K\in Comp(Y))
\\
&
(C(F,K)\doteqdot
\sup_{(x,s)\in X\times K}
\|F(x)(s)\|_{B(Z)}<\infty)
\}
\\
{\mathbf{M}}_{x}
&
\doteqdot{\overline}{\{
F(x)\mid F\in{\mathcal{M}}
\}}
\end{aligned}$$ Let denote by ${\mathfrak{V}}\doteqdot{\left\langle {\mathfrak{E}},\pi,X{\right\rangle}}$ the trivial bundle with constant stalk $Z$ so $\Gamma(\pi)
\simeq{\mathcal{C}_{b}
\left(X,Z\right)}$, set $$\label{18252606}
\begin{cases}
{\mathcal{A}}_{x}
\doteqdot\{\mu_{(v,x)}^{K}
\mid
K\in Comp(Y),
v\in\Gamma(\pi)
\},
\\
\mu_{(v,x)}^{K}:
{\mathbf{M}}_{x}\ni G\mapsto
\sup_{s\in K}
\|G(s)v(x)\|,
\\
{\mathbf{M}}
\doteqdot\{
{\left\langle {\mathbf{M}}_{x},{\mathcal{A}}_{x}{\right\rangle}}
\}_{x\in X}.
\end{cases}$$ Then by using Lemma \[22312406\] and the cited [@gie Theorem $5.9$] we can construct a bundle of $\Omega-$space say ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ whose stalk at $x$ is the locally convex space ${\left\langle {\mathbf{M}}_{x},{\mathcal{A}}_{x}{\right\rangle}}$ and whose space of bounded continuous sections $\Gamma(\pi_{{\mathbf{M}}})$ is such that $\Gamma(\pi_{{\mathbf{M}}})\simeq{\mathcal{M}}$.

Let $f\in\prod_{x\in\ X}{\mathbf{M}}_{x}$ be such that $(\forall K\in Comp(Y))
(\sup_{(x,s)\in X\times K}
\|f(x)(s)\|_{B(Z)}<\infty)$ then (see Thm. \[22372406\]) $(1)
\Leftrightarrow
(2)
\Leftrightarrow
(3)$ with

1.  $(\forall K\in Comp(Y))
    (\forall v\in\Gamma(\pi))$ $$(\lim_{x\to x_{\infty}}
    \sup_{s\in K}
    \|
    f(x)(s)v(x)
    -
    f(x_{\infty})(s)v(x)
    \|
    =0);$$

2.  $f\in\Gamma^{x_{\infty}}(\pi_{{\mathbf{M}}})$;

3.  $f:X\to{\mathcal{C}_{c}
    \left(Y,B_{s}(Z)\right)}$ continuous at $x_{\infty}$.

Moreover (see Thm. \[22372406\]) if $Y$ is locally compact for all $t\in Y$ $$\label{17150703}
\Gamma(\pi_{{\mathbf{M}}})_{t}
\bullet
\Gamma(\pi)
\subseteq
\Gamma(\pi).$$ *Therefore we constructed two bundles ${\mathfrak{V}}\doteqdot{\left\langle {\mathfrak{E}},\pi,X{\right\rangle}}$ and ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ whose topologies are $(I)$ stalkwise related by $\{{\mathcal{A}}_{x}\}_{x\in X}$ in and for which hold $(1)\Leftrightarrow(2)$ and $(II)$ globally related by . Finally $\Gamma^{x_{\infty}}(\pi_{{\mathbf{M}}})$ coincide with the subset of all maps $f:X\to{\mathcal{C}_{c}
\left(Y,B_{s}(Z)\right)}$ continuous at $x_{\infty}$ such that $(\forall K\in Comp(Y))
(\sup_{(x,s)\in X\times K}
\|f(x)(s)\|_{B(Z)}<\infty)$.* The natural generalization of the mentioned property $(I)$ leads to the concept of **$\left(\Theta,{\mathcal{E}}\right)-$structure**, see Definition \[10282712\] and Lemma \[15482712\], while the generalization of the property $(II)$ leads to that of **compatible** $\left(\Theta,{\mathcal{E}}\right)-$structure, see Definition \[10282712\].

A similar and more important global correlation between ${\mathfrak{M}}$ and ${\mathfrak{E}}$, this time for the case in which the topology on each stalk ${\mathfrak{M}}_{x}$ is that of the pointwise convergence instead of the compact convergence, is that encoded in in the definition of “invariant” $\left(\Theta,{\mathcal{E}},\mu\right)-$structures, see Definition \[17161902\], for a similar definition see Definition \[10282712\]. This conclude the discussion about the relationship between the topologies on ${\mathfrak{M}}$ and ${\mathfrak{E}}$, in particular between those on ${\mathfrak{B}}$ and ${\mathfrak{E}}$ [^2]

 Kurtz’s General Approximation Theorem  {#kurtzs-general-approximation-theorem .unnumbered}
---------------------------------------

Briefly I shall recall what here has to be understood as a classical stability problem in order to better explain how to generalize it through the language of bundles. The classical stability problem could be so described: fixed a Banach space $Z$ find a sequence $\{S_{n}:D_{n}\subseteq Z\to Z\}$ of possibly unbounded linear operators in $Z$ and a sequence $\{P_{n}\}\subset B(Z)$ where $P_{n}$ is a continuous spectral projector of $S_{n}$ for $n\in{\mathbb{N}}$, such that if

$(A)$

:   there exists an operator $S:D\subset Z\to Z$ such that $S=\lim_{n\to\infty}S_{n}$ with respect to a suitable topology or in any other generalized sense,

$(B)$

:   such that there exists a spectral projector $P\in B(Z)$ of $S$ such that $P=\lim_{n\to\infty}P_{n}$ with respect to the strong operator topology.

Here a spectral projector of an operator $S$ in a Banach space is a continuous projector associated to a closed $S-$invariant subspace $Z_{0}$ such that $
\sigma(S{\upharpoonright}Z_{0})
\subset
\sigma(S)
$, where $\sigma(T)$ is the spectrum of the operator $T$.

In Ch $IV$ [@kato] there are many stability theorems in which the previous limit of operators $S_{n}$ has to be understood with respect to the metric induced by the “gap” between the corresponding closed graphs.

Moreover there are stability theorems even for operators defined in different spaces, obtained by using the concept of *Transition Operators* introduced by Victor Burenkov, see for expample [@bl1], [@bl2] and [@bll], or the results obtained by Massimo Lanza de Cristoforis and Pier Domenico Lamberti by using functional analytic approaches, see for examples [@l1], [@l2], [@ll].

If one try to generalize the classical stability problem to the case in which $Z$ is replaced by any sequence $\{Z_{n}\}$ of Banach spaces and $S_{n}$ is defined in $Z_{n}$ for all $n$, then he would face the following difficulty: how to adapt the definition of the gap given by Kato to the case of a sequence of different spaces, more in general in which sense to understand the convergence of operators defined in different spaces.

A first step toward the generalization to the case of different spaces of the classical stability problem is the following result Thomas G. Kurtz, [@kurtz].

\[$2.1.$ of [@kurtz]\] \[16250603\] For each $n$, let $U_{n}(t)$ be a strongly continuous contraction semigroup defined on $L_{n}$ with the infinitesimal operator $A_{n}$. Let $A=ex-\lim_{n\to\infty}A_{n}$. Then there exists a strongly continuous semigroup $U(t)$ on $L$ such that $\lim_{n\to\infty}U_{n}(t)Q_{n}f=U(t)f$ for all $f\in L$ and $t\in{\mathbb {R}}^{+}$ if and only if the domain $D(A)$ is dense and the range $R(\lambda_{0}-A)$ of $\lambda_{0}-A$ is dense in $L$ for some $\lambda_{0}>0$. If the above conditions hold $A$ is the infinitesimal generator of $U$ and we have $$\label{19100603}
\lim_{n\to\infty}\sup_{0\leq s\leq t}
\|
U_{n}(s)Q_{n}f-
Q_{n}U(s)f
\|_{n}=0,$$ for every $f\in L$ and $t\in{\mathbb {R}}^{+}$.

Here ${\left\langle L,\|\cdot\|{\right\rangle}}$ is a Banach space, $\{{\left\langle L_{n},\|\cdot\|_{n}{\right\rangle}}\}_{n\in{\mathbb{N}}}$ is a sequence of Banach spaces, $\{Q_{n}\in B(L,L_{n})\}_{n\in{\mathbb{N}}}$ such that $\lim_{n\to\infty}\|Q_{n}f\|_{n}=\|f\|$ for all $f\in L$. Let $f\in L$ and $\{f_{n}\}_{n\in{\mathbb{N}}}$ such that $f_{n}\in L_{n}$ for every $n\in{\mathbb{N}}$, thus he set [^3] $$\label{16590603}
f=\lim_{n\to\infty}f_{n}
\Leftrightarrow
\lim_{n\to\infty}
\|f_{n}-Q_{n}f\|_{n}=0.$$ Moreover if $A_{n}:Dom(A_{n})\subseteq L_{n}\to L_{n}$ he defined $$\label{16500603}
\begin{cases}
Graph(ex-lim_{n\to\infty}A_{n})
\doteqdot
\{
\lim_{n\in{\mathbb{N}}}
s_{0}(n)
\mid
s_{0}\in\Phi_{0}
\}
\\
\begin{aligned}
&
\Phi_{0}
\doteqdot
\{
(f_{n},A_{n}f_{n})_{n\in{\mathbb{N}}}
\in
(Z\times Z)^{{\mathbb{N}}}
\mid
\\
&
(\forall n\in{\mathbb{N}})
(f_{n}\in Dom(A_{n}))
\wedge
(\exists\,
\lim_{n\in{\mathbb{N}}}(
f_{n},A_{n}f_{n}))
\},
\end{aligned}
\end{cases}
\tag{Gr}$$ where $(f,g)=\lim_{n\in{\mathbb{N}}}
(f_{n},A_{n}f_{n}))
$ iff $f=\lim_{n\in{\mathbb{N}}}f_{n}$ and $g=\lim_{n\in{\mathbb{N}}}A_{n}f_{n}$ and all these limits are those defined in . Whenever $Graph(ex-lim_{n\to\infty}A_{n})$ is a graph in $L$ he denoted by $ex-lim_{n\to\infty}A_{n}$ the corresponding operator in $L$.

The Kurtz’s approach, just now described, did not make use of the bundle theory, and, except when imposing stronger assumptions, it cannot be implemented in a bundle of $\Omega-$spaces’ approach.

The following consideration results fundamental for understanding the strategy behind this work. There is a strong resemblance of with . More importantly **if the topology on ${\mathfrak{M}}$ and that on ${\mathfrak{E}}$ are related by a $\left(\Theta,{\mathcal{E}}\right)-$structure (for a simple example see ) then there exists a “resemblance” of the selection convergence with the convergence of the sequence of semigroups $\{U_{n}\}_{n\in{\mathbb{N}}}$ to the semigroup $U$**. [^4]

I used the word resemble due to the difficulty to build a couple of reasonable Kurtz’ bundles, i.e. two bundles of $\Omega-$spaces ${\left\langle {\mathfrak{E}},\pi,X{\right\rangle}}$ and ${\left\langle {\mathfrak{M}},\rho,X{\right\rangle}}$ such that $X$ is the Alexandroff compactification of ${\mathbb{N}}$ and , hold. In any case it is possible under strong assumptions, see Setion \[17572301\].

Although the difficulty of constructing a couple of Kurtz’s bundles, *the recognition of the before mentioned two resemblances, were sufficient to push me in investigating the possibility of extending the Kurtz’s Theorem \[16250603\] in the general framework of bundles of $\Omega-$spaces, by using $\left(\Theta,{\mathcal{E}}\right)-$structure.*

 Direct sum of Bundles of $\Omega-$spaces and (Pre)Graph Sections  {#direct-sum-of-bundles-of-omega-spaces-and-pregraph-sections .unnumbered}
------------------------------------------------------------------

It should now be clear that in the way of extending the Kurtz’s Theorem I am replacing the sequence of Banach spaces $\{L_{n}\}_{n\in{\mathbb{N}}\cup\{\infty\}}$ and $\{{\mathcal{C}_{c}
\left({\mathbb {R}}^{+},B_{s}(L_{n)}\right)}
\}_{n\in{\mathbb{N}}\cup\{\infty\}}$, where $L_{\infty}\doteqdot L$, with a Banach bundle ${\mathfrak{E}}$ and the bundle of $\Omega-$spaces ${\mathfrak{M}}$ respectively, while the Kurtz’ convergences and with convergences of selections on the bundles spaces ${\mathfrak{M}}$ and ${\mathfrak{E}}$ respectively. In this view definition has to be replaced by that of Pre-Graph sections, see Definition \[16161212bis\] (essentially ), while the case in which $Graph(ex-lim_{n\to\infty}A_{n})$ is a graph in $L$ with that of Graph sections, see Definition \[12432110bis\]. Hence it arises as a natural question which topology has to be selected for the bundle space of ${\mathfrak{V}}\oplus{\mathfrak{V}}$.

An essential tool used in the definition of $Graph(ex-lim_{n\to\infty}A_{n})$ in is that of convergence of a sequence $(f_{n},A_{n}f_{n})$ in the direct sum of the spaces $L_{n}\oplus L_{n}$, given by construction as the convergence of both the sequences in $L_{n}$ in the meaning of .

It is exactly this factorization property the property which I want to preserve when selecting the “right” topology on the bundle space of ${\mathfrak{V}}\oplus{\mathfrak{V}}$.

It is a well-known result the solution of this problem in the special case of Banach bundles. I generalize this result for a finite direct sum of bundles of $\Omega-$spaces, by constructing in Theorem \[16322110\] a family of seminorms on the direct sum of Hausdorff locally convex spaces which is fundamental for any of the following equivalent topologies: the direct sum top. the $lc$-direct sum top., the box top. and most importantly the product topology.

The result that the fundamental set is direct along with Lemma \[19420111\] allow to define the direct sum of bundles of $\Omega-$spaces as given in Definition \[12422110\].

Finally the fundamental result that the topology on each stalk is the product topology, (fact encripted in ) the choice given in of the subset of section of the direct sum of bundles and the general convergence criterium in , allow to show the claimed factorization property in Corollary \[17571212\]. I.e. any selection of the direct sum $\bigoplus_{i=1}^{n}{\mathfrak{E}}_{i}$ of bundles is continuous at a point if and only if is continuous at the same point its projection selection of ${\mathfrak{E}}_{i}$ for every $i=1,...,n$.

 Semigroup Approximation Theorem.  {#semigroup-approximation-theorem. .unnumbered}
----------------------------------

Roughly speaking ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}\in
\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$ iff ${\mathcal{T}}(x)$ is the graph of the infinitesimal generator $T_{x}$ of a $C_{0}-$semigroup ${\mathcal{U}}(x)$ on ${\mathfrak{E}}_{x}$, for all $x\in X$, holds and $${\mathcal{U}}\in\Gamma^{x_{\infty}}(\rho).$$ Thus, according the discussed way of extending the Kurtz’ theorem which I intend to perform in this work, to find an element in the class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$ means to find an extension of Theorem \[16250603\]. In the first main result of this work, **Theorem \[17301812b\]**, has been constructed an element of the class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$.

 Laplace Duality Properties  {#laplace-duality-properties .unnumbered}
----------------------------

There are essentially two strong hypothesis to be satisfied in Theorem \[17301812b\]. In constructing a model for hypothesis $[ii]$ one obtains **Corollary \[21343012\]**. In any case the most important one is hypothesis $[i]$, i.e. the assumption that the $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the Laplace duality property, see Definition \[16401812b\].

Roughly speaking the full Laplace duality property means that the natural action of $\prod_{x\in X}
{\mathcal{L}}({\mathfrak{E}}_{x})
$ over $\prod_{x\in X}{\mathfrak{E}}_{x}$, induces, by restriction, an action over $\Gamma(\pi)$ of the Laplace trasform of $\Gamma(\rho)$. More exactly $$\label{Laplace}
\tag{${\mathbf{LD}}$}
\begin{cases}
(\forall\lambda>0)
\left({\mathfrak{L}}(\Gamma(\rho))
(\cdot)(\lambda)
\bullet
\Gamma(\pi)
\subseteq
\Gamma(\pi)\right)
\\
{\mathfrak{L}}(F)(x)(\lambda)
\doteqdot
\int_{0}^{\infty}
e^{-\lambda s}
F(x)(s)\,ds
\doteq
\int_{{\mathbb {R}}^{+}}
F(x)(s)\,d\mu_{\lambda}(s).
\end{cases}$$

See Def. \[15531102\], Def. \[15062301\] and Def. \[16401812b\]. The implicit assumption is that for all $x\in X$ and $\lambda>0$ $${\mathfrak{M}}_{x}
\subseteq
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},
{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}),\mu_{\lambda}),$$ where $\mu_{\lambda}$ is the Laplace measure associated to $\lambda$ and ${\mathfrak{L}}_{1}({\mathbb {R}}^{+},
{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}),\mu_{\lambda})$ is the space of all $\mu_{\lambda}-$integrable maps with values in the locally convex space ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$. Among others informations in Prop. \[18390901\] (an application of a result in [@SilInt]) there are reasonable conditions ensuring the previous inclusion.

I investigated in Section \[13320803\] a strategy for constructing classes having the full Laplace duality property. Although I worked in this section in a wide generality, here I shall present the applications of the results in the case of interest for the present introduction.

Firstly we note that by construction $$\Gamma(\pi)
\subset
\prod_{x\in X}{\mathfrak{E}}_{x},$$ hence the most natural (duality) action to consider over $\Gamma(\pi)$ is the restriction on it of the “standard” [^5] action of $${\mathcal{L}}\left(\prod_{x\in X}{\mathfrak{E}}_{x}\right).$$ Secondly we recall that the Laplace duality property is described in terms of the action restricted over $\Gamma(\pi)$ of a subspace of $\prod_{x\in X}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},
{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\mu_{\lambda})$.

Therefore the idea is to construct a suitable **locally convex space ${\mathfrak{G}}$** and a linear map $\Psi$ such that $$\label{18250803}
\begin{cases}
{\mathfrak{G}}
\subset
{\mathcal{L}}\left(\prod_{x\in X}{\mathfrak{E}}_{x}\right)
\,
\text{ as linear spaces}
\\
\Psi:
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})
\to
\prod_{x\in X}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},
{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\mu_{\lambda}),
\end{cases}$$ and most importantly such that the following relation between the two actions holds for all ${\overline}{F}
\in{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$, $x\in X$, $\lambda>0$ and $v\in\Gamma(\pi)$ $$\label{18100803}
{\left\langle \int
\Psi({\overline}{F})(x)(s)
\,d\mu_{\lambda}(s),v(x){\right\rangle}}_{x}
=
{\left\langle \int{\overline}{F}(s)\,d\mu_{\lambda}(s),v{\right\rangle}}(x),$$ see **Corollary \[15111901\]**. Here ${\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ is the space of all $\mu_{\lambda}-$integrable maps on ${\mathbb {R}}^{+}$ and at values in the locally convex space ${\mathfrak{G}}$, while for any linear space $E$ we denote by ${\left\langle \cdot,\cdot{\right\rangle}}:
End(E)
\times
E
\to
E$ the standard duality.

**It is exactly by that we can rewrite as a duality problem**. More exactly if $\exists\,{\mathcal{F}}\subset
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi({\mathcal{F}})=\Gamma(\rho)$ then $$\label{17280803a}
{\mathbf{LD}}
\Leftrightarrow
(\forall\lambda>0)
({\left\langle {\mathcal{A}}_{\lambda},\Gamma(\pi){\right\rangle}}
\subseteq\Gamma(\pi)),$$ where for all $\lambda>0$ $$\label{17280803b}
{\mathcal{A}}_{\lambda}
\doteqdot
\left\{
\int{\overline}{F}(s)\,d\mu_{\lambda}(s)
\mid{\overline}{F}\in{\mathcal{F}}
\right\}
\subset
{\mathcal{L}}\left(\prod_{x\in X}{\mathfrak{E}}_{x}\right).$$ *There are two advantage of decoding the problem of finding the full Laplace duality property into the invariance problem . Firstly is a classical problem of invariance of a subspace of a linear topological space for the standard action of a subspace of the space of all linear continuous operators on it. Secondly in has involved through the definition of ${\mathcal{A}}_{\lambda}$ the space ${\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$, while in has involved the much more complicate space $\prod_{x\in X}
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},
{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\mu_{\lambda})$ indeed $\Gamma(\rho)$ belongs to it.*

The crucial idea behind Definition \[10221801\] of the space ${\mathfrak{G}}$ is the use of the concept of **locally convex final topology**. Indeed the well-known property of this topology allows in Lemma \[11011501\], to ensure that for all $v\in\Gamma(\pi)$ the evaluation map $$\label{18050803}
{\mathfrak{G}}\ni
A\mapsto A v
\in
\prod_{x\in X}{\mathfrak{E}}_{x}\,
\text{ is continuous.}$$ And is essentially a consequence of . Although we are mainly interested to the equality , there is an important result strictly determined by the locally convex final topology on ${\mathfrak{G}}$. Namely Theorem \[15251401\] ensures that holds the second statement in and that for all ${\overline}{F}\in{\mathfrak{L}}_{1}
({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ $$\int\Pr_{x}(\Psi({\overline}{F}))(s)
\,d\mu_{\lambda}(s)
=
\Pr_{x}
\circ
\left[
\int{\overline}{F}(s)\,d\mu_{\lambda}(s)
\right]
\circ\imath_{x}.$$

 The class $\Delta
{\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$.\
Projection Approximation Theorem.  {#the-class-deltaleftlangle-mathfrakvmathfrakdthetamathcalerightrangle.-projection-approximation-theorem. .unnumbered}
-----------------------------------------------------------------------------------

In this section we shall discuss the main result of this work namely **Theorem \[13020103\]**, ensuring the existence of an element ${\left\langle {\mathcal{T}},\Phi,x_{\infty}{\right\rangle}}$ of the class $\Delta
{\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$ such that $T_{x}$ is the infinitesimal generator of a $C_{0}-$semigroup of contractions on ${\mathfrak{E}}_{x}$ and such that there exists a selection $$\label{17321002}
\boxed{
{\mathcal{P}}
\in
\Gamma^{x_{\infty}}(\eta),}$$ satisfying and such that ${\mathcal{P}}(x)$ is a spectral projector of $T_{x}$ for all $x\in X$, where $T_{x}$ is the operator whose graph is ${\mathcal{T}}(x)$.

We shall start with the definition of the class $\Delta
{\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$, given in Definition \[15312011bis\]. Roughly speaking given a $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ and denoted ${\mathfrak{D}}
\doteqdot
{\left\langle {\mathfrak{B}},\eta,X{\right\rangle}}$, we say that ${\left\langle {\mathcal{T}},\Phi,x_{\infty}{\right\rangle}}
\in
\Delta
{\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$ iff for all $x\in X$ the set ${\mathcal{T}}(x)$ is a graph in ${\mathfrak{E}}_{x}$, holds and there exists a selection ${\mathcal{P}}
\in
\Gamma^{x_{\infty}}(\eta)$ continuous at $x_{\infty}$ such that ${\mathcal{P}}(x)$ is a projector on ${\mathfrak{E}}_{x}$ and for all $x\in X$ $$\label{20020803}
{\mathcal{P}}(x)
T_{x}
\subseteq
T_{x}
{\mathcal{P}}(x),$$ where $T_{x}$ is the operator in ${\mathfrak{E}}_{x}$ whose graph is ${\mathcal{T}}(x)$.

In others words ${\left\langle {\mathcal{T}},\Phi,x_{\infty}{\right\rangle}}
\in
\Delta
{\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$ iff ${\mathcal{T}}$ is a selection of graphs in ${\mathfrak{E}}$ continuous at $x_{\infty}$ in the meaning of and such that there exists a selection ${\mathcal{P}}$ of projectors on ${\mathfrak{E}}$ continuous at $x_{\infty}$ such that *${\mathcal{P}}$ commutes with ${\mathcal{T}}$* in the meaning of .

Notice that implies for all $x\in X$ that the following $$T_{x}
{\mathcal{P}}(x)
Dom(T_{x})
\subseteq
{\mathcal{P}}(x){\mathfrak{E}}_{x}$$ is satisfied by any spectral projector $P_{x}$ of the opearor $T_{x}$, indeed by definition for them we have $T_{x}
P_{x}
{\mathfrak{E}}_{x}
\subseteq
P_{x}{\mathfrak{E}}_{x}$. Viceversa whenever $T_{x}$ is the infinitesimal generator of a $C_{0}-$semigroup ${\mathcal{W}}_{T}(x)$ of contractions on ${\mathfrak{E}}_{x}$, the most important case in this work, it results that is the property satisfied by all the spectral projectors of the form $${\mathcal{P}}(x)
\doteqdot
-
\frac{1}{2\pi i}
\int_{\Gamma}
R(-T_{x};\zeta)\,
d\zeta,$$ where $R(-T_{x};\zeta)$ is the resolvent map of the operator $-T_{x}$ and $\Gamma$ is a suitable closed curve on the complex plane. Hence we can consider the commutation in as the defining property of what we here consider as the “interesting” bundle ${\mathcal{P}}$ of spectral projectors associated to ${\mathcal{T}}$.

###  Proof of Theorem \[13020103\]. *Invariant* $\left(\Theta,{\mathcal{E}},\mu\right)-$ structures.  {#proof-of-theorem-13020103.-invariant-leftthetamathcalemuright--structures. .unnumbered}

Let us describe the principle steps and new structures required on proving Theorem \[13020103\]. The first property involved in showing is that of an invariant $\left(\Theta,{\mathcal{E}}\right)-$structure, see Definition \[10282712\]. The characteristich property of an invariant $\left(\Theta,{\mathcal{E}}\right)-$ structure ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ is the following one $$\label{17481003}
\left\{
F\in\prod_{z\in X}^{b}
{\mathfrak{M}}_{z}
\mid
(\forall t\in Y)
(F_{t}
\bullet
{\mathcal{E}}(\Theta)
\subseteq
\Gamma(\pi))
\right\}
=
\Gamma(\rho),$$ where ${\mathfrak{W}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ and $F_{t}(x)\doteqdot F(x)(t)$.

The first reason of intoducing the concept of invariant structure is that the global relation implies the following corresponding local one for any $x_{\infty}\in X$, see Lemma \[14452602\] $$\label{17481003loc}
\left\{
F\in\prod_{z\in X}{\mathfrak{M}}_{z}
\mid
F
\bullet
\Gamma^{x_{\infty}}(\pi)
\subseteq
\Gamma^{x_{\infty}}(\pi)
\right\}
\subseteq
\Gamma^{x_{\infty}}(\rho).$$ Hence if we show that $$\label{18271003}
{\mathcal{P}}
\bullet
\Gamma^{x_{\infty}}(\pi)
\subseteq
\Gamma^{x_{\infty}}(\pi)$$ the follows by for the special case $Y=\{pt\}$.

The presence of the uniform convergence over compact subsets of $Y$ may drastically restrict the class of invariant $\left(\Theta,{\mathcal{E}}\right)-$ structures. In order to deal with this problem we introduced the concept of $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,{\mathbb {R}}^{+}{\right\rangle}}$, with ${\mathfrak{Q}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{H}},\gamma{\right\rangle}},\xi,X,{\mathfrak{R}}{\right\rangle}}$, Definition \[17161902\]. It is essentially the same definition, given for $\left(\Theta,{\mathcal{E}}\right)-$ structures but with the following differences $$\label{20081003}
\begin{cases}
{\mathfrak{H}}_{x}
\subseteq
{\mathfrak{L}}_{1}
\left(Y,{\mathcal{L}}_{S_{x}}
({\mathfrak{E}}_{x}),\mu\right)
\\
{\mathfrak{H}}_{x}
\text{ with the pointwise topology}.
\end{cases}$$ Thus on each stalk ${\mathfrak{H}}_{x}$ the topology is that of pointwise convergence on $Y$ instead of that of compact convergence.

Thus we have an *invariant* $\left(\Theta,{\mathcal{E}},\mu\right)-$ structures iff: $$\label{14451103}
\left\{
F\in\prod_{z\in X}{\mathfrak{H}}_{z}
\mid
(\forall t\in Y)
(F_{t}
\bullet
{\mathcal{E}}(\Theta)
\subseteq
\Gamma(\pi))
\right\}
=
\Gamma(\xi),$$ The second reason to be interested to the the invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure, resides in the fact that it is a necessary assumption in order to have the following implication, see Corollary \[21152602\] and Remark \[12500303\] $$\label{17401003}
{\mathcal{W}}_{T}
\in
\Gamma^{x_{\infty}}(\xi)
\Rightarrow
{\mathcal{P}}
\bullet
\Gamma^{x_{\infty}}(\pi)
\subseteq
\Gamma^{x_{\infty}}(\pi).$$ But as we know the first main result Theorem \[17301812b\] states that $
{\mathcal{W}}_{T}
\in
\Gamma^{x_{\infty}}(\rho)$ for a $\left(\Theta,{\mathcal{E}}\right)-$ structure which we know to be not a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure.

Hence we need a way of connecting the two types of structures. This is performed by the definition of the $\left(\Theta,{\mathcal{E}},\mu\right)-$structure ${\left\langle {\mathfrak{V}},
{\mathfrak{V}}({\mathbf{M}}^{\mu},\Gamma(\rho)),X,Y{\right\rangle}}$ underlying ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$, see Def, \[18072802\].

In view of the property which we are looking for, namely , we have to maintain the vicinity of the original and the underlying structure. This is performed by using [@gie Theorem $5.9$] for constructing bundles with a given subspace of continuous sections. In this way the space $\Gamma(\rho)$ of continuous sections of ${\mathfrak{W}}$ will be a subspace of the space $\Gamma
(\pi_{{\mathbf{M}}^{\mu}})$ of all continuous sections of the underlying bundle ${\mathfrak{V}}({\mathbf{M}}^{\mu},\Gamma(\rho))$, (the equality if $X$ is compact).

Thus we have, see Proposition \[23332802\], $$\label{17501003}
\Gamma^{x_{\infty}}(\rho)
\subseteq
\Gamma^{x_{\infty}}
(\pi_{{\mathbf{M}}^{\mu}}).$$ Finally by Theorem \[17301812b\] we know that ${\mathcal{W}}_{T}
\in\Gamma^{x_{\infty}}(\rho)$ therefore , and , follows by and . In addition to invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structures, for obtaining we need another concept, namely that of a $\mu-$related couple ${\left\langle {\mathfrak{V}},{\mathfrak{Z}}{\right\rangle}}$, Definition \[15492502\] and a bundle type generalization of the Lebesgue Theorem, see Theorem \[15101701\].

 Summary of the main results ans structures  {#summary-of-the-main-results-ans-structures .unnumbered}
============================================

The main results of this work are the following ones

1.  An explicit construction of a direct fundamental set of seminorms of the topological direct sum of a finite family of Hausdorff locally convex spaces, (Theorem \[16322110\]);

2.  Characterization of selections of ${\mathfrak{W}}$ continuous at a point when ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ is a $\left(\Theta,{\mathcal{E}}\right)-$structure, (Lemma \[15482712\]);

3.  Construction of a $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ and characterization of a subclass of $\Gamma^{x_{\infty}}(\rho)$ when ${\mathfrak{V}}$ is trivial, (Theorem \[22372406\])

4.  Construction of an element in the class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$, (**Theorem \[17301812b\]**, Corollary \[21343012\]);

5.  Conditions in order to satisfy the bounded equicontinuity of which in hypothesis $(ii)$ of Theorem \[17301812b\] (Corollary \[21343012\]);

6.  Conditions in order to have (Proposition \[18390901\]);

7.  The technical Lemma \[11011501\] and Theorem \[15251401\];

8.  **Theorem \[15332203\]** and Corollaries \[15111901\] and \[18491004\];

9.  ${\mathcal{K}}-$Uniform Convergence **Theorem \[10581004\]**;

10. Consequence of being an ${\left\langle \nu,\eta,E,Z,T{\right\rangle}}$ invariant set $V$ with respect to ${\mathcal{F}}$ (Proposition \[18310302\]);

11. Construction of a class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$ by using an ${\left\langle \nu,\eta,{\mathfrak{G}},K(\Gamma),{\mathbb {R}}^{+}{\right\rangle}}$ invariant set $V$ with respect to $\{{\overline}{F}_{T}\}$ (Corollary \[13511102\]);

12. A bundle version of the Lebesgue theorem for a $\mu-$related couple ${\left\langle {\mathfrak{V}},{\mathfrak{Z}}{\right\rangle}}$ (Theroem \[15101701\]);

13. Technical Lemmas \[12151902\] and \[14452602\];

14. Corollary \[15262502\]

15. Construction of a selection of spectral projectors continuous at a point given a selection of semigroups continuous at the same point (Corollary \[21152602\] and Remark \[12500303\]);

16. The Main result of this work is the construction of an element in the class $\Delta
    {\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$ (**Theorem \[13020103\]**).

The main structures defined in this works are the following ones

1.  Direct sum of bundles of $\Omega-$spaces (Definition \[12422110\]);

2.  (Invariant) $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$, (**Definition \[10282712\]**);

3.  Graph section ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}$, (**Definition \[12432110bis\]**);

4.  Class $\Delta{\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$, (**Definition \[15312011bis\]**);

5.  Class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$, (**Definition \[19490412bis\]**);

6.  Class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$;

7.  ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ with the Laplace duality property, (Definition \[16401812b\]);

8.  ${\mathbf{U}}-$Spaces (**Definition \[14302503\]**);

9.  The locally convex space ${\mathfrak{G}}$ (**Definition \[10221801\]**);

10. ${\left\langle \nu,\eta,E,Z,T{\right\rangle}}$ invariant set $V$ with respect to ${\mathcal{F}}$ (Definition \[20030202\]);

11. $\mu-$related couple ${\left\langle {\mathfrak{V}},{\mathfrak{Z}}{\right\rangle}}$ (Definition \[15492502\]);

12. (Invariant) $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ (Definition \[17161902\]);

13. $\left(\Theta,{\mathcal{E}}\right)-$ structure ${\left\langle {\mathfrak{V}},{\mathfrak{V}}({\mathbf{M}},\Gamma(\xi)),X,Y{\right\rangle}}$ underlying a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ (**Definition \[18072802\]**).

Direct Sum
==========

Notations {#notat}
---------

Let $E$ be a topological vector space, ${\left\langle {\mathcal{L}}(E),\tau{\right\rangle}}$ the linear space of all continuous linear maps on $E$ with the topology $\tau$ compatible with the linear structure. Thus ${\mathbf{U}}({\left\langle {\mathcal{L}}(E),\tau{\right\rangle}})$ is the class of all continuous semigroup morphisms defined on ${\mathbb {R}}^{+}$ and with values in ${\mathcal{L}}_{\tau}(E)$, moreover if $\|\cdot\|$ is any seminorm on ${\mathcal{L}}(E)$ (not necessarly continuous with respect to $\tau$) we set ${\mathbf{U}}_{\|\cdot\|}
({\left\langle {\mathcal{L}}(E),\tau{\right\rangle}})$ as the subset of all $U\in{\mathbf{U}}({\left\langle {\mathcal{L}}(E),\tau{\right\rangle}})$ such that $\|U(s)\|\leq 1$, for all $s\in{\mathbb {R}}^{+}$. Finally we set ${\mathbf{U}}_{is}
({\left\langle {\mathcal{L}}(E),\tau{\right\rangle}})$ as the subset of all ${\mathbf{U}}({\left\langle {\mathcal{L}}(E),\tau{\right\rangle}})$ such that there exists a fundamental set $\Gamma$ of seminorms on $E$ such that $U(s)$ is an isometry with respect to any element in $\Gamma$, for all $s\in{\mathbb {R}}^{+}$. We use throughout this work the notations of [@gie] and often when referring to Banach bundles those of [@fell]. In particular $
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},p,X,{\mathfrak{N}}{\right\rangle}}
$ or simply $
{\left\langle {\mathfrak{E}},p,X{\right\rangle}}
$, whenever $\tau$ and ${\mathfrak{N}}$ are known, is a bundle of $\Omega-$spaces ($1.5.$ of [@gie]), where we denote by $\tau$ the topology on ${\mathfrak{E}}$ while with $
{\mathfrak{N}}\doteqdot
\{
\nu_{j}\mid
j\in J
\}
$ the directed set of seminorms on ${\mathfrak{E}}$ ($1.3.$ of [@gie]). Thus we set $
{\mathfrak{N}}_{x}
\doteqdot
\{
\nu_{j}^{x}
\mid
j\in J
\}
$ with $
\nu_{j}^{x}
\doteqdot
\nu_{j}{\upharpoonright}{\mathfrak{E}}_{x}
$ for all $x\in X$ and $j\in J$. Moreover for any $U\subseteq X$ we shall call the *Space of Sections of ${\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},p,X,{\mathfrak{N}}{\right\rangle}}$ on $U$* the linear space $\Gamma_{U}(p)$ of all continuous bounded selections of $p$ defined on $U$, namely [^6] $$\Gamma_{U}(p)
\doteqdot
{\mathcal{C}_{}
\left(U,{\mathfrak{E}}\right)}
\bigcap
\prod_{x\in U}^{b}
{\left\langle {\mathfrak{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}$$ where $
{\mathfrak{E}}_{x}
\doteqdot
\overset{-1}{p}(x)
$, $$\prod_{x\in U}^{b}
{\left\langle {\mathfrak{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}
\doteqdot
\left\{
\sigma\in
\prod_{x\in U}
{\mathfrak{E}}_{x}
\mid
\sup_{x\in U}
\nu_{j}^{x}
(\sigma(x))
<\infty
\right\},$$ where $
{\mathcal{C}_{}
\left(U,{\mathfrak{E}}\right)}
$ is the linear space of all continuous maps $f:U\to{\mathfrak{E}}$. Let $U\subseteq X$ and $x\in U$ set $$\Gamma_{U}^{x}(p)
\doteqdot
\left\{
f\in
\prod_{x\in U}^{b}
{\left\langle {\mathfrak{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}
\mid
f\text{ is continuous at $x$}
\right\}.$$ So $\Gamma_{U}(p)
=
\bigcap_{x\in U}
\Gamma_{U}^{x}(p)$. We set $\Gamma(p)
\doteqdot
\Gamma_{X}(p)$ and $\Gamma^{x}(p)
\doteqdot
\Gamma_{X}^{x}(p)$ for any $x\in X$. The definition of trivial bundle of $\Omega-$spaces is given in $1.8.$ of [@gie].

All vector spaces are assumed to be over ${\mathbb{K}}\in\{{\mathbb {R}},{\mathbb {C}}\}$, Hlcs is for Hausdorff locally convex spaces. We say that $
{\mathbf{V}}
\doteqdot
\{{\left\langle V_{x},{\mathcal{A}}_{x}{\right\rangle}}\}_{x\in X}
$ is a *nice* family of Hlcs if $\{V_{x}\}_{x\in X}$ is a family of Hlcs such that $\exists\,J$ for which $\forall x\in X$ the set $
{\mathcal{A}}_{x}
\doteqdot
\{
\mu_{j}^{x}
\}_{j\in J}
$ is a directed [^7] family of seminorms on $V_{x}$ generating the lct on it. For any family of seminorms $\Gamma$ on a vector space $V$ we shall define the *directed family of seminorms associated to $\Gamma$* the following set $
\{
\sup F
\mid
F\in{\mathcal{P}}_{\omega}(\Gamma)
\}
$ with the standard order relation of pointwise order on $
{\mathbb {R}}^{V}
$.

Given two lcs $E$ and $F$ we denote by $
{\mathcal{L}}(E,F)
$ the linear space of all linear and continuous maps on $E$ with values in $F$, and set $
{\mathcal{L}}(E)
\doteqdot
{\mathcal{L}}(E,E)
$, moreover by $
\Pr(E)
\doteqdot
\{
P\in{\mathcal{L}}(E)
\mid
P\circ P
=P
\}
$ we denote the class of all projectors on $E$. Let $S$ be a class of bounded subsets of a lcs $E$, thus $
{\mathcal{L}}_{S}(E)
$ denotes the lcs whose underlining linear space is ${\mathcal{L}}(E)$ and whose lct is that of uniform convergence over the subsets in $S$. When $E$ is a normed space and $S$ is the class of all finite parts of $E$, then ${\mathcal{L}}_{S}(E)$ will be denoted by $B_{s}(E)$, while $B(E)$ denotes ${\mathcal{L}}(E)$ with the usual norm topology.

Let $X,Y$ be two topological spaces then ${\mathcal{C}_{}
\left(X,Y\right)}$ is the set of all continuous maps on $X$ valued in $Y$, while ${\mathcal{C}_{c}
\left(X,Y\right)}$ is the topological space of all continuous maps on $X$ valued in $Y$ with the topology of uniform convergence over the compact subsets of $Y$. If $Y$ is a uniform space then ${\mathcal{C}}^{b}(X,Y)$ is the space of all bounded maps in ${\mathcal{C}_{}
\left(X,Y\right)}$, while ${\mathcal{C}}_{c}^{b}(X,Y)
=
{\mathcal{C}_{c}
\left(X,Y\right)}\cap
{\mathcal{C}}^{b}(X,Y)$. If $E$ is a lcs then ${\mathcal{C}_{c}
\left(X,E\right)}$ is a lcs, while if $E$ is a Hlcs and $Comp(X)$ is a covering of $X$, for example if $X$ is a locally compact space, then ${\mathcal{C}_{c}
\left(X,E\right)}$ is a Hlcs.

Let $Y$ be a locally compact space, $\mu\in Radon(Y)$ and $E\in Hlcs$, then ${\mathfrak{L}}_{1}(Y,E,\mu)$ denotes the linear space of all scalarly essentially $\mu-$integrable maps $f:Y\to E$ such that its integral belongs to $E$, see [@IntBourb Ch. $6$], while $Meas(Y,{\mathfrak{E}}_{x},\mu)$ denotes the linear space of all $\mu-$measurable maps $f:Y\to E$.

If $S$ is any set then ${\mathcal{P}}_{\omega}(S)$ denotes the class of all finite subsets of $S$. Finally “u.s.c.” is for upper semicontinuous. Finally we shall give the following

\[15012602\] Let $
{\mathfrak{A}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{B}},\tau{\right\rangle}},\xi,X,{\mathfrak{N}}{\right\rangle}}
$ be a bundle of $\Omega-$spaces, $x\in X$ and $Q,S$ subsets of $\prod_{z\in X}{\mathfrak{B}}_{z}$. Thus we set $$Q_{S}^{x}
\doteqdot
\{
H\in Q\mid
(\exists\,F\in S)
(H(x)=F(x))
\},$$ and $Q_{\diamond}^{x}
\doteqdot
Q_{\Gamma(\xi)}^{x}$, while $\Gamma_{S}^{x}(\xi)
\doteqdot
(\Gamma^{x}(\xi))_{S}^{x}$ and $\Gamma_{\diamond}^{x}(\xi)
\doteqdot
(\Gamma^{x}(\xi))_{\diamond}^{x}$.

 Direct Sum of Bundles $\Omega-$spaces 
---------------------------------------

###  Standard construction of Bundles of $\Omega-$spaces 

\[ $FM(3)-FM(4)$ ([@gie] $\S 5$) and $FM(3^{*})-FM(4)$ \] \[17471910s\] Let $
{\mathbf{V}}
\doteqdot
\{{\left\langle V_{x},{\mathcal{A}}_{x}{\right\rangle}}\}_{x\in X}
$ be a nice family of Hlcs with $
{\mathcal{A}}_{x}
\doteqdot
\{\mu_{j}^{x}
\}_{j\in J}
$ for all $x\in X$; we say that ${\mathcal{G}}$ satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{V}}$ (see [@gie] $\S 5$) if $
{\mathcal{G}}
\subseteq
\prod_{x\in X}^{\infty}
{\left\langle V_{x},{\mathcal{A}}_{x}{\right\rangle}}
\doteqdot
\left\{
f\in
\prod_{x\in X}
V_{x}
\mid
(\forall j\in J)
(\sup_{x\in X}
\mu_{j}^{x}(f(x))
<\infty)
\right\}
$ and

$FM(3)$

:   $
    \{f(x)\mid f\in {\mathcal{G}}\}
    $ is dense in $V_{x}$ for all $x\in X$;

$FM(4)$

:   $
    X\ni x
    \mapsto
    \mu_{j}^{x}(f(x))
    $ is u.s.c. $\forall j\in J$ and $\forall f\in{\mathcal{G}}$.

Now we shall introduce a stronger condition namely we say that $
{\mathcal{G}}
$ satisfies $FM(3^{*})-FM(4)$ with respect to ${\mathbf{V}}$ if $FM(3^{*})$ and $FM(4)$ hold where $$\label{15012310}
(\forall x\in X)
(\{f(x)\mid f\in {\mathcal{G}}\}
=
V_{x}).
\tag*{$FM(3^{*})$}$$

\[21132406\] Let $
{\mathbf{V}}_{k}
\doteqdot
\{{\left\langle V_{x},{\mathcal{A}}_{x}^{k}{\right\rangle}}\}_{x\in X}$, with $k=1,2$, be two nice families of Hlcs such that ${\mathcal{A}}_{x}^{1}$ and ${\mathcal{A}}_{x}^{2}$ generate the **same** locally convex topology on $V_{x}$. Notice that ${\mathcal{G}}$ satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{V}}_{1}$ **doesn’t imply** that ${\mathcal{G}}$ satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{V}}_{2}$. Say ${\mathcal{A}}^{k}=\{\mu^{j_{k}}\mid j_{k}\in J_{k}\}$, then a necessary condition for which the fact described happens is the following one $(\exists\, X_{0}\subseteq X)
(\exists\, k\in\{1,2\})
(\exists j_{s}:X_{0}\times J_{k}
\to J_{s})
(\forall x\in X_{0})
(\forall j_{k}\in J_{k})
(\exists\,C>0)
(\forall v\in V_{x})$ $$\mu_{j_{k}}^{x}(v)\leq C
\mu_{j_{s}(x,j_{k})}^{x}(v).$$ In other words the index for some inequalities relating the seminorms on ${\mathcal{A}}^{k}$ with those on ${\mathcal{A}}^{s}$, depends on $x$.

\[18111610\] Let $
{\mathbf{V}}'
\doteqdot
\{{\left\langle V_{x},{\mathcal{A}}_{x}'{\right\rangle}}\}_{x\in X}
$ be a family of Hlcs where $
{\mathcal{A}}_{x}'
\doteqdot
\{\mu_{j_{x}}^{x}
\}_{j_{x}\in J_{x}}
$ is a directed family of seminorms on $V_{x}$ generating the lct on it, for all $x\in X$. Then we set $$\begin{cases}
J
\doteqdot
\bigcup_{x\in X}\{x\}\times J_{x},
\\
\mu_{(y,j_{y})}^{x}
\doteqdot
\begin{cases}
\mu_{j_{x}}^{x},\,
x=y
\\
{\mathbf{0}},\,
x\neq y
\end{cases}
\forall x,y\in X,
\forall j_{y}\in J_{y},
\\
{\mathcal{A}}_{x}
\doteqdot
\{\mu_{j}^{x}
\}_{j\in J},\,
\forall x\in X.
\end{cases}$$

Moreover by setting $$\prod_{x\in X}^{\infty}
{\left\langle V_{x},{\mathcal{A}}_{x}'{\right\rangle}}
\doteqdot
\left\{
f\in
\prod_{x\in X}
V_{x}
\mid
\left(
\forall {\overline}{j}\in\prod_{x\in X}J_{x}
\right)
\left(\sup_{x\in X}
\mu_{{\overline}{j}(x)}^{x}(f(x))
<\infty\right)
\right\},$$ we say that $
{\mathcal{G}}
\subseteq
\prod_{x\in X}^{\infty}
{\left\langle V_{x},{\mathcal{A}}_{x}'{\right\rangle}}
$ satisfies $FM(3)-FM(4')$ with respect to ${\mathbf{V}}'$ if $FM(3)$ and $FM(4')$ hold where $$\label{17311811}
\left(\forall{\overline}{j}\in J\right)
\left(\forall f\in{\mathcal{G}}\right)
\left(X\ni x
\mapsto
\mu_{{\overline}{j}(x)}^{x}(f(x))
\text{is u.s.c.}\right).
\tag*{$FM(4')$}$$

\[18111744\] Definition \[18111610\] ensures the possibility of associating a nice family of Hlcs to any family of Hlcs. Namely let $
{\mathbf{V}}'
\doteqdot
\{{\left\langle V_{x},{\mathcal{A}}_{x}'{\right\rangle}}\}_{x\in X}
$ be a family of Hlcs where $
{\mathcal{A}}_{x}'
\doteqdot
\{\mu_{j_{x}}^{x}
\}_{j_{x}\in J_{x}}
$ is a directed family of seminorms on $V_{x}$ generating the lct on it, for all $x\in X$. Then $
{\mathbf{V}}
\doteqdot
\{{\left\langle V_{x},{\mathcal{A}}_{x}{\right\rangle}}\}_{x\in X}
$ is a nice family of Hlcs, called the *nice family of Hlcs associated to ${\mathbf{V}}'$*. Indeed ${\mathcal{A}}_{x}$ generates the lct on $V_{x}$ moreover it is trivially directed. Moreover $$\prod_{x\in X}^{\infty}
{\left\langle V_{x},{\mathcal{A}}_{x}'{\right\rangle}}
=
\prod_{x\in X}^{\infty}
{\left\langle V_{x},{\mathcal{A}}_{x}{\right\rangle}}.$$ and $
{\mathcal{G}}
$ satisfies $FM(3)-FM(4')$ with respect to ${\mathbf{V}}'$ if and only if it satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{V}}$.

\[$5.2-5.3$ of [@gie]\] \[17471910Ba\] Let $
{\mathbf{E}}
=
\{
{\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}
\}_{x\in X}
$ be a nice family of Hlcs with $
{\mathfrak{N}}_{x}
\doteqdot
\{\nu^{x}_{j}\mid j\in J\}
$ for all $x\in X$. Moreover let ${\mathcal{E}}$ satisfy $FM(3)-FM(4)$ with respect to $
{\mathbf{E}}
$. Now we shall apply to ${\mathbf{E}}$ and ${\mathcal{E}}$ the general procedure described in $5.2-5.3$ of [@gie] for constructing bundles of $\Omega-$spaces.

We define $${\mathfrak{V}}({\mathbf{E}},{\mathcal{E}})$$ to be the *Bundle generated by the couple ${\left\langle {\mathbf{E}},{\mathcal{E}}{\right\rangle}}$* , if

1.  $
    {\mathfrak{V}}({\mathbf{E}},{\mathcal{E}})
    =
    {\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}),\tau({\mathbf{E}},{\mathcal{E}}){\right\rangle}},\pi_{{\mathbf{E}}},X,{\mathfrak{N}}{\right\rangle}}
    $;

2.  $
    {\mathfrak{E}}({\mathbf{E}})
    \doteqdot
    \bigcup_{x\in X}
    \{x\}
    \times
    {\mathbf{E}}_{x}
    $, $
    \pi_{{\mathbf{E}}}:{\mathfrak{E}}({\mathbf{E}})
    \ni
    (x,v)
    \mapsto x
    \in X
    $.

3.  $
    {\mathfrak{N}}=\{\nu_{j}\mid j\in J\}
    $, with $
    \nu_{j}:{\mathfrak{E}}
    \ni
    (x,v)
    \mapsto
    \nu_{j}^{x}(v)
    $;

4.  $
    \tau({\mathbf{E}},{\mathcal{E}})
    $ is the topology on ${\mathfrak{E}}$ [^8] such that for all $
    (x,v)\in
    {\mathfrak{E}}({\mathbf{E}})
    $ $${\mathcal{I}}_{(x,v)}^{\tau({\mathbf{E}},{\mathcal{E}})}
    \doteqdot
    {\mathfrak{F}_{{\mathcal{B}}_{{\mathbf{E}}}((x,v))}^{{\mathfrak{E}}({\mathbf{E}})}}$$ is the neighbourhood’s filter of $(x,v)$ with respect to it. Here ${\mathfrak{F}_{{\mathcal{B}}((x,v))}^{{\mathfrak{E}}({\mathbf{E}})}}$ is the filter on ${\mathfrak{E}}({\mathbf{E}})$ generated by the following filter’s base $$\begin{aligned}
    {1}
    {\mathcal{B}}_{{\mathbf{E}}}((x,v))
    \doteqdot
    \{
    T_{{\mathbf{E}}}(U,\sigma,{\varepsilon},j)
    &
    \mid
    U\in Open(X),
    \sigma\in{\mathcal{E}},
    {\varepsilon}>0,
    j\in J
    \\
    &
    \mid
    x\in U,
    \nu_{j}^{x}(v-\sigma(x))
    <{\varepsilon}\},\end{aligned}$$ where $$\label{14332310a}
    T_{{\mathbf{E}}}(U,\sigma,{\varepsilon},j)
    \doteqdot
    \left\{
    (y,w)\in{\mathfrak{E}}({\mathbf{E}})
    \mid
    y\in U,
    \nu_{j}^{y}(w-\sigma(y))
    <{\varepsilon}\right\};$$

What is important in this construction is the fact that ${\mathcal{E}}$ is canonically isomorphic to a linear subspace of $\Gamma(\pi_{{\mathbf{E}}})$ indeed

\[17150312\] Let $
{\mathbf{E}}
=
\{
{\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}
\}_{x\in X}
$ be a nice family of Hlcs with $
{\mathfrak{N}}_{x}
\doteqdot
\{\nu^{x}_{j}\mid j\in J\}
$ for all $x\in X$. Moreover let ${\mathcal{E}}$ satisfy $FM(3)-FM(4)$ with respect to $
{\mathbf{E}}
$, and ${\mathfrak{V}}({\mathbf{E}},{\mathcal{E}})$ be the bundle generated by the couple $
{\left\langle {\mathbf{E}},{\mathcal{E}}{\right\rangle}}
$. Thus according Prop. $5.8$ of [@gie] we have

1.  $
    {\mathfrak{V}}({\mathbf{E}},{\mathcal{E}})
    $ is a bundle of $\Omega-$spaces;

2.  with the notations of Definition \[17471910Ba\] ${\mathfrak{V}}({\mathbf{E}},{\mathcal{E}})$ is such that

    1.  ${\left\langle {\mathfrak{E}}({\mathbf{E}})_{x},\tau({\mathbf{E}},{\mathcal{E}}){\right\rangle}}$ as topological vector space is isomorphic to ${\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}$ for all $
        x\in X
        $;

    2.  $
        {\mathcal{E}}
        $ is canonically isomorphic [^9] to a linear subspace of $\Gamma(\pi_{{\mathbf{E}}})$ and if $X$ is compact and ${\mathbf{E}}$ is a function module, see [@gie 5.1.], then $
        {\mathcal{E}}
        \simeq
        \Gamma(\pi_{{\mathbf{E}}})
        $.

\[15412610\] Let ${\mathbf{E}}$ be a nice family of Hlcs and let ${\mathcal{E}}$ satisfy $FM(3-4)$ with respect to ${\mathbf{E}}$. Thus for all $U\in Open(X)$, $\sigma\in{\mathcal{E}}$, ${\varepsilon}>0$, $j\in J$ $$T_{{\mathbf{E}}}(U,\sigma,{\varepsilon},j)
=
\bigcup_{y\in U}
B_{{\mathbf{E}}_{y},j,{\varepsilon}}(\sigma(y))$$ where for all $
s
\in
{\mathbf{E}}_{y}
$ $$B_{{\mathbf{E}}_{y},j,{\varepsilon}}(s)
\doteqdot
\left\{
(y,w)
\in{\mathfrak{E}}({\mathbf{E}})_{y}
\mid
\nu_{j}^{y}
\left(w-s\right)
<{\varepsilon}\right\}.$$ In others words $T_{{\mathbf{E}}}(U,\sigma,{\varepsilon},j)$ is the $\sigma-$ deformed cilinder of radius ${\varepsilon}$, which justifies the name of ${\varepsilon}-$tube.

###  Characterizations of Neighbourhood’s filters and Sections of Bundles of $\Omega-$spaces 

The following are simple but very useful characterizations of the convergence and of a section in a bundle of $\Omega-$spaces.

\[28111555\] Let ${\mathfrak{V}}
=
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ be a bundle of $\Omega-$spaces where $
{\mathfrak{N}}\doteqdot
\{
\nu_{j}\mid
j\in J
\}
$. Moreover let $b\in{\mathfrak{E}}$ and $\{b_{\alpha}\}_{\alpha\in D}$ a net in ${\mathfrak{E}}$. Then $(1)
\Leftarrow
(2)
\Leftarrow
(3)
\Leftrightarrow
(4)$ where

1.  $\lim_{\alpha\in D}b_{\alpha}=b$;

2.  $
    (\exists\,U\in Op(X)\mid U\ni\pi(b))
    (\exists\,\sigma\in\Gamma_{U}(\pi))
    (\sigma\circ\pi(b)=b)
    $ such that $ 
    \lim_{\alpha\in D}\pi(b_{\alpha})
    =\pi(b)
    $ and $
    (\forall j\in J)
    (\lim_{\alpha\in D}
    \nu_{j}(b_{\alpha}-\sigma(\pi(b_{\alpha})))
    =0)
    $;

3.  $
    (\exists\,U'\in Op(X)\mid U'\ni\pi(b))
    (\exists\,\sigma'\in\Gamma_{U}(\pi)\mid
    \sigma'\circ\pi(b)=b)
    $ and $
    (\forall U\in Op(X)\mid U\ni\pi(b))
    (\forall\sigma\in\Gamma_{U}(\pi)\mid
    \sigma\circ\pi(b)=b)
    $ we have $ 
    \lim_{\alpha\in D}\pi(b_{\alpha})
    =\pi(b)
    $ and $
    (\forall j\in J)
    (\lim_{\alpha\in D}
    \nu_{j}(b_{\alpha}-\sigma(\pi(b_{\alpha})))
    =0)
    $;

4.  $
    (\exists\,U'\in Op(X)\mid U'\ni\pi(b))
    (\exists\,\sigma'\in\Gamma_{U}(\pi))
    (\sigma'\circ\pi(b)=b)
    $ and $\lim_{\alpha\in D}b_{\alpha}=b$.

Moreover if ${\mathfrak{V}}$ is locally full then $(1)\Leftrightarrow(4)$.

Of course $(3)\rightarrow(2)$. $(2)$ is equivalent to say that $
(\exists\,U\in Op(X)\mid U\ni\pi(b))
(\exists\,\sigma\in\Gamma_{U}(\pi))
(\sigma\circ\pi(b)=b)
$ such that $
(\forall V\in Op(X)\mid 
\pi(b)\in
V\subseteq U)
(\exists\,\alpha(V)\in D)
(\forall\alpha\geq\alpha(V))
(\pi(b_{\alpha})
\in V)
$ and $
(\forall j\in J)
(\forall{\varepsilon}>0)
(\exists\,\alpha(V)\in D)
(\forall\alpha\geq\alpha(j,{\varepsilon}))
(\nu_{j}(b_{\alpha}-\sigma(\pi(b_{\alpha})))
<{\varepsilon})
$. Set $
\alpha(V,j,{\varepsilon})\in D
$ such that $
\alpha(V,j,{\varepsilon})
\geq
\alpha(V),
\alpha(j,{\varepsilon})
$ which there exists $D$ being directed, thus we have $
(\forall V\in Op(X)\mid 
\pi(b)\in
V\subseteq U)
(\forall j\in J)
(\forall{\varepsilon}>0)
(\exists\,\alpha(V,j,{\varepsilon})\in D)
$ such that $(\forall\alpha\geq\alpha(V,j,{\varepsilon}))
(\nu_{j}(b_{\alpha}-\sigma(\pi(b_{\alpha})))
<{\varepsilon})$ and $\pi(b_{\alpha})
\in V$. Thus $(1)$ follows by applying $1.5.\,VII$ of [@gie]. Finally by applying $1.5.\,VII$ of [@gie] $(4)$ (respectively $(1)$ if ${\mathfrak{V}}$ is locally full) is equivalent to $
(\exists\,U'\in Op(X)\mid U'\ni\pi(b))
(\exists\,\sigma'\in\Gamma_{U}(\pi))
(\sigma'\circ\pi(b)=b)
$ and $
(\forall\sigma\in\Gamma_{U}(\pi)\mid
\sigma\circ\pi(b)=b)
(\forall j\in J)
(\forall{\varepsilon}>0)
(\forall V\in Op(X)\mid 
\pi(b)\in
V\subseteq U)
(\exists\,{\overline}{\alpha}\in D)
(\forall\alpha\geq{\overline}{\alpha})
$ we have $
\pi(b_{\alpha})
\in V
$ and $
\nu_{j}(b_{\alpha}-\sigma(\pi(b_{\alpha})))
<{\varepsilon}$ which is $(3)$.

Although the following is a simple consequence of the previous result, we give to it the status of Theorem due to its extraordinary importance and use in the whole this work.

\[15380512\] Let ${\mathfrak{V}}=
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ be a bundle of $\Omega-$spaces, $W\subseteq X$ and indicate $
{\mathfrak{N}}=
\{
\nu_{j}
\mid
j\in J
\}
$. Moreover let $f\in{\mathfrak{E}}^{W}$, $x_{\infty}\in W$. Then $
(1)
\Leftarrow
(2)
\Leftrightarrow
(3)
\Leftarrow
(4)
\Leftrightarrow
(5)
$ where

1.  $f$ is continuous in $x_{\infty}$;

2.  $
    (\exists\,U\in Op(X)\mid U\ni x_{\infty})
    (\exists\,\sigma\in\Gamma_{U}(\pi))
    (\sigma(x_{\infty})
    =f(x_{\infty}))$ such that $
    \nu_{j}\circ(f-\sigma\circ\pi\circ f):
    W\cap U\to{\mathbb {R}}$ and $\pi\circ f:W\to X$ are continuous in $x_{\infty}$ for all $j\in J$;

3.  $\pi\circ f:W\to X$ is continuous in $x_{\infty}$ and $
    (\exists\,U\in Op(X)\mid U\ni x_{\infty})
    (\exists\,\sigma\in\Gamma_{U}(\pi))
    (\sigma(x_{\infty})
    =f(x_{\infty}))$ such that $$(\forall j\in J)
    (\lim_{
    y\to x_{\infty},y\in W\cap U}
    \nu_{j}(f(y)-\sigma
    \circ\pi\circ f(y))=0);$$

4.  $
    (\exists\,U'\in Op(X)\mid U'\ni x_{\infty})
    (\exists\,\sigma'\in\Gamma_{U}(\pi))
    (\sigma'(x_{\infty})
    =f(x_{\infty}))$ and $
    (\forall U\in Op(X)\mid U\ni x_{\infty})
    (\forall\sigma\in\Gamma_{U}(\pi)
    \mid
    \sigma(x_{\infty})
    =f(x_{\infty}))$ we have $
    \nu_{j}\circ(f-\sigma):
    W\cap U\to{\mathbb {R}}$ and $\pi\circ f:W\to X$ are continuous in $x_{\infty}$ for all $j\in J$;

5.  $\pi\circ f:W\to X$ is continuous in $x_{\infty}$ and $
    (\exists\,U'\in Op(X)\mid U'\ni x_{\infty})
    (\exists\,\sigma'\in\Gamma_{U}(\pi))
    (\sigma'(x_{\infty})
    =f(x_{\infty}))$ and $
    (\forall U\in Op(X)\mid U\ni x_{\infty})
    (\forall\sigma\in\Gamma_{U}(\pi)
    \mid
    \sigma(x_{\infty})
    =f(x_{\infty}))$ we have $$(\forall j\in J)
    (\lim_{
    y\to x_{\infty},y\in W\cap U}
    \nu_{j}(f(y)-\sigma\circ\pi\circ f(y))=0);$$

6.  $
    (\exists\,U'\in Op(X)\mid U'\ni x_{\infty})
    (\exists\,\sigma'\in\Gamma_{U}(\pi))
    (\sigma'(x_{\infty})
    =f(x_{\infty}))$ and $f$ is continuous at $x_{\infty}$.

Moreover if ${\mathfrak{V}}$ is locally full then $(1)\Leftrightarrow(6)$ and if it is full we can choose $U=X$ and $U'=X$.

$(1)$ is equivalent to say that for each net $
\{x_{\alpha}\}_{\alpha\in D}
\subset
W
$ such that $\lim_{\alpha\in D}x_{\alpha}= x_{\infty}$ in $W$, we have $\lim_{\alpha\in D}f(x_{\alpha})
=f( x_{\infty})$ in ${\mathfrak{E}}$. Similarly $(2)$ is equivalent to say that for each net $
\{x_{\alpha}\}_{\alpha\in D}
\subset
W
$ such that $\lim_{\alpha\in D}x_{\alpha}= x_{\infty}$ in $W$, we have $
\lim_{\alpha\in D}\pi\circ f(x_{\alpha})
=
\pi\circ f(x_{\infty})
$ and $
(\forall j\in J)
(\lim_{\alpha\in D}
\nu_{j}\circ(f-\sigma\circ\pi\circ f)
(x_{\alpha})
=
\nu_{j}\circ(f-\sigma\circ\pi\circ f)
(x_{\infty}))
$. Thus $(1)
\Leftarrow
(2)$ follows by the corresponding one in Proposition \[28111555\] with the positions $
(\forall\alpha\in D)
(b_{\alpha}\doteqdot f(x_{\alpha}))
$ and $b\doteqdot f(x_{\infty})$. Similarly $(1)\Leftarrow(5)$ follows by $(1)\Leftarrow(3)$ of Proposition \[28111555\]. Finally $(5)\Rightarrow(6)$ follows by $(5)\Rightarrow(1)$, while if $(6)$ is true then $\pi\circ f$ is continuous at $x_{\infty}$ indeed $\pi$ is continuous, then $(5)$ follows by the implication $(4)\Rightarrow(3)$ of Proposition \[28111555\] with the positions $
(\forall\alpha\in D)
(b_{\alpha}\doteqdot f(x_{\alpha}))
$ and $b\doteqdot f(x_{\infty})$.

\[28111707\] Let ${\mathfrak{V}}
=
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ be a bundle of $\Omega-$spaces, $W\subseteq X$ and indicate $
{\mathfrak{N}}=
\{
\nu_{j}\mid
j\in J
\}
$. Moreover let $f\in\prod_{x\in W}{\mathfrak{E}}_{x}$ and $x_{\infty}\in W$. Then $
(1)
\Leftarrow
(2)
\Leftrightarrow
(3)
\Leftarrow
(4)
\Leftrightarrow
(5)
\Leftrightarrow
(6)
$ where

1.  $f$ is continuous in $x_{\infty}$;

2.  $(\exists\,U\in Op(X)\mid U\ni x_{\infty})
    (\exists\,\sigma\in\Gamma_{U}(\pi))
    (\sigma(x_{\infty})
    =f(x_{\infty}))$ such that $
    \nu_{j}\circ(f-\sigma):
    W\cap U\to{\mathbb {R}}$ is continuous in $x_{\infty}$ for all $j\in J$;

3.  $(\exists\,U\in Op(X)\mid U\ni x_{\infty})
    (\exists\,\sigma\in\Gamma_{U}(\pi))
    (\sigma(x_{\infty})
    =f(x_{\infty}))$ such that $$(\forall j\in J)
    (\lim_{
    y\to x_{\infty},y\in W\cap U}
    \nu_{j}(f(y)-\sigma(y))=0);$$

4.  $(\exists\,U'\in Op(X)\mid U'\ni x_{\infty})
    (\exists\,\sigma'\in\Gamma_{U}(\pi))
    (\sigma'(x_{\infty})
    =f(x_{\infty}))$ and $
    (\forall U\in Op(X)\mid U\ni x_{\infty})
    (\forall\sigma\in\Gamma_{U}(\pi)
    \mid
    \sigma(x_{\infty})
    =f(x_{\infty}))$ we have that $
    \nu_{j}\circ(f-\sigma):
    W\cap U\to{\mathbb {R}}$ is continuous in $x_{\infty}$ for all $j\in J$;

5.  $(\exists\,U'\in Op(X)\mid U'\ni x_{\infty})
    (\exists\,\sigma'\in\Gamma_{U}(\pi))
    (\sigma'(x_{\infty})
    =f(x_{\infty}))$ and $
    (\forall U\in Op(X)\mid U\ni x_{\infty})
    (\forall\sigma\in\Gamma_{U}(\pi)
    \mid
    \sigma(x_{\infty})
    =f(x_{\infty}))$ we have $$(\forall j\in J)
    (\lim_{
    y\to x_{\infty},y\in W\cap U}
    \nu_{j}(f(y)-\sigma(y))=0).$$

6.  $(\exists\,U'\in Op(X)\mid U'\ni x_{\infty})
    (\exists\,\sigma'\in\Gamma_{U}(\pi))
    (\sigma'(x_{\infty})
    =f(x_{\infty}))$ and $f$ is continuous at $x_{\infty}$

If ${\mathfrak{V}}$ is locally full then $(1)
\Leftrightarrow
(6)$ and if it is full we can choose $U=X$ and $U'=X$.

By Theorem \[15380512\] and $\pi\circ f=Id$.

\[16572003\] Let ${\mathfrak{V}}$ be full and such that there exists a linear space $E$ such that for all $x\in X$ there exists a linear subspace $E_{x}\subseteq E$ such that ${\mathfrak{E}}_{x}
=
\{x\}\times E_{x}$, and that [^10] $$\{
X\ni x
\mapsto
(x,v)\in{\mathfrak{E}}_{x}
\mid v\in
\bigcap_{x\in X}
E_{x}
\}
\subset
\Gamma(\pi),$$ If $f_{0}\in\prod_{x\in X}E_{x}$ and

$f\in\prod_{x\in X}{\mathfrak{E}}_{x}$ such that $f(x)=(x,f_{0}(x))$ for all $x\in X$ and $f_{0}(x_{\infty})\in
\bigcap_{x\in X}
E_{x}$, then $(1)
\Leftrightarrow(2)
\Leftrightarrow(3)$, where

1.  $f$ is continuous at $x_{\infty}$

2.  $(\exists\,U\in Op(X)\mid U\ni x_{\infty})
    (\exists\,\sigma\in{\mathcal{C}_{b}
    \left(U,E\right)})
    (\sigma(x_{\infty})
    =f(x_{\infty}))$ such that for all $j\in J$ $$\lim_{z\to x_{\infty},z\in W\cap U}
    \nu_{j}^{z}(f(z)-\sigma(z))=0;$$

3.  for all $j\in J$ $$\lim_{z\to x_{\infty},z\in W\cap U}
    \nu_{j}^{z}((z,f_{0}(z))-
    (z,f(x_{\infty})))=0.$$

By Corollary \[28111707\] $(1)\Leftrightarrow(2)$. Let $(3)$ hold then $(2)$ is true by setting $\sigma=\tau_{f(x_{\infty})}{\upharpoonright}U$. Let $(2)$ hold then $
\nu_{j}^{z}((z,f_{0}(z))-
(z,f(x_{\infty})))
\leq
\nu_{j}^{z}((z,f_{0}(z))-
\sigma(z))
+
\nu_{j}^{z}(\sigma(z)-
\tau_{f(x_{\infty})}(z))$, thus $(3)$ follows by $(2)$ and by Corollary \[28111707\] applied to the continuous map $\tau_{f(x_{\infty})}{\upharpoonright}U$.

\[21492812\] Let ${\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ be a bundle of $\Omega-$spaces, $W\subseteq X$ and indicate $
{\mathfrak{N}}=
\{
\nu_{j}\mid
j\in J
\}
$. Moreover let $f,g\in\prod_{x\in W}{\mathfrak{E}}_{x}$ and $x_{\infty}\in W$. Then if ${\mathfrak{V}}$ locally full or $\nu_{j}$ is continuous $\forall j\in J$, then $
(1)
\rightarrow
(2)
$ where

1.  $f(x_{\infty})=g(x_{\infty})$ and $f$ and $g$ are continuous in $x_{\infty}$;

2.  $
    (\exists\,U\in Op(X)\mid
    x_{\infty}\in U)
    $ such that $$(\forall j\in J)
    (\lim_{
    y\to x_{\infty},y\in W\cap U}
    \nu_{j}(f(y)-g(y))=0).$$

Moreover if ${\mathfrak{V}}$ is full we can choose $U=X$.

The statement is trivial in the case of continuiuty of all the $\nu_{j}$. Whereas if ${\mathfrak{V}}$ is locally full by $(1)\rightarrow(5)$ of Corollary \[28111707\] we have $(\exists\,U\in Op(X))
(\exists\,\sigma\in\Gamma_{U}(\pi))
(\sigma(x_{\infty})
=f(x_{\infty})
=g(x_{\infty}))$ such that $$(\forall j\in J)
(\lim_{y\to x_{\infty},y\in W\cap U}
\nu_{j}(f(y)-\sigma(y))
=
\lim_{
y\to x_{\infty},y\in W\cap U}
\nu_{j}(g(y)-\sigma(y))
=0).$$ Therefore $$\lim_{y\to x_{\infty},y\in W\cap U}
\nu_{j}(f(y)-g(y))
\leq
\lim_{y\to x_{\infty},y\in W\cap U}
\nu_{j}(f(y)-\sigma(y))
+
\lim_{y\to x_{\infty},y\in W\cap U}
\nu_{j}(g(y)-\sigma(y))
=0.$$

\[281117010\] Let $
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}
$ be a bundle of $\Omega-$spaces, $W\in Op(X)$ and indicate $
{\mathfrak{N}}=
\{
\nu_{j}\mid
j\in J
\}
$. Moreover let $f\in\prod_{x\in W}^{b}{\mathfrak{E}}_{x}$. Then $
(1)
\Leftarrow
(2)
\Leftrightarrow
(3)
\Leftarrow
(4)
\Leftrightarrow
(5)
$ where

1.  $f\in\Gamma_{W}(\pi)$;

2.  $$(\forall x\in\ W)
    (\exists\,U_{x}\in Op(X)\mid U_{x}\ni x)
    (\exists\,\sigma_{x}\in\Gamma_{U_{x}}(\pi))
    (\sigma_{x}(x)
    =f(x))$$ such that $
    \nu_{j}
    \circ(f-\sigma_{x})
    $ is continuous in $x$, $\forall j\in J$;

3.  $$(\forall x\in\ W)
    (\exists\,U_{x}\in Op(X)\mid U_{x}\ni x)
    (\exists\,\sigma_{x}\in\Gamma_{U_{x}}(\pi))
    (\sigma_{x}(x)
    =f(x))$$ such that $
    (\forall j\in J)
    (\lim_{
    y\to x,y\in W\cap U_{x}}
    \nu_{j}(f(y)-\sigma_{x}(y))
    =
    0)$;

4.  $$(\forall x\in\ W)
    (\exists\,U_{x}'\in Op(X)\mid U_{x}'\ni x)
    (\exists\,\sigma_{x}'\in\Gamma_{U_{x}}(\pi))
    (\sigma_{x}'(x)
    =f(x))$$ and $$(\forall\,U_{x}\in Op(X)\mid U_{x}\ni x)
    (\forall\sigma_{x}\in\Gamma_{U_{x}}(\pi)
    \mid
    \sigma_{x}(x)
    =f(x))$$ we have that $
    \nu_{j}
    \circ(f-\sigma_{x})
    $ is continuous in $x$ for all $x\in W$ and $j\in J$;

5.  $$(\forall x\in\ W)
    (\exists\,U_{x}'\in Op(X)\mid U_{x}'\ni x)
    (\exists\,\sigma_{x}'\in\Gamma_{U_{x}}(\pi))
    (\sigma_{x}'(x)
    =f(x))$$ and $$(\forall x\in\ W)
    (\forall\,U_{x}\in Op(X)\mid U_{x}\ni x)
    (\forall\sigma_{x}\in\Gamma_{U_{x}}(\pi)
    \mid
    \sigma_{x}(x)
    =f(x))$$ we have $(\forall j\in J)
    (\lim_{
    y\to x,y\in W\cap U_{x}}
    \nu_{j}(f(y)-\sigma_{x}(y))
    =
    0)$.

By Corollary \[28111707\].

In the case in which the bundle is locally full we can give some useful characterizations of the Neighbourhood’s filter ${\mathcal{I}}_{\alpha}^{\tau}$ of any point $\alpha$ in the bundle space ${\left\langle {\mathfrak{E}},\tau{\right\rangle}}$.

\[${\varepsilon}-$Tubes\] \[12592310\] Let $
{\mathfrak{P}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},p,X,{\mathfrak{N}}{\right\rangle}}
$ be a locally full bundle of $\Omega-$spaces, and let us denote $
{\mathfrak{N}}\doteqdot
\{
\nu_{j}
\mid
j\in J
\}
$. Set $$\begin{cases}
{\mathcal{K}}^{loc}
\doteqdot
\prod_{\alpha\in{\mathfrak{E}}}
{\mathcal{K}}_{\alpha}^{loc}
\\
{\mathcal{K}}_{\alpha}^{loc}
\doteqdot
\left\{
(U,\sigma_{U})
\mid
U\in Op(X),
\sigma_{U}\in\Gamma_{U}(p)
\mid
p(\alpha)\in U,
\sigma_{U}(p(\alpha))=\alpha
\right\}.
\end{cases}$$ Moreover $\forall\alpha\in{\mathfrak{E}}$ and $\forall{\mathfrak{l}}\in{\mathcal{K}}^{loc}$ set $$\begin{cases}
{\mathcal{B}}_{{\mathfrak{l}}}^{loc}(\alpha)
\doteqdot
\left\{
T^{loc}
(V,{\mathfrak{l}}_{2}(\alpha),{\varepsilon},j)
\mid
V\in Op(X),
{\varepsilon}>0,
j\in J
\mid
p(\alpha)
\in V
\subseteq
{\mathfrak{l}}_{1}(\alpha)
\right\},
\\
T^{loc}(U,\sigma_{U},{\varepsilon},j)
\doteqdot
\left\{
\beta\in{\mathfrak{E}}
\mid
p(\beta)\in U,
\nu_{j}(\beta-\sigma_{U}(p(\beta)))
<{\varepsilon}\right\},
\end{cases}$$ $
(\forall U\in Op(X))
(\forall j\in J)
(\forall{\varepsilon}>0)
(\forall\sigma_{U}\in\Gamma_{U}(p))
$.

If $
{\mathfrak{P}}
$ is a full bundle then we can set $$\begin{cases}
{\mathcal{K}}
\doteqdot
\prod_{\alpha\in{\mathfrak{E}}}
{\mathcal{K}}_{\alpha}
\\
{\mathcal{K}}_{\alpha}
\doteqdot
\left\{
(U,\sigma)
\mid
U\in Op(X),
\sigma\in\Gamma(p)
\mid
p(\alpha)\in U,
\sigma(p(\alpha))=\alpha
\right\}.
\end{cases}$$ Moreover $\forall\alpha\in{\mathfrak{E}}$ and $\forall{\mathfrak{l}}\in{\mathcal{K}}$ set $$\begin{cases}
{\mathcal{B}}_{{\mathfrak{l}}}(\alpha)
\doteqdot
\left\{
T(V,{\mathfrak{l}}_{2}(\alpha),{\varepsilon},j)
\mid
V\in Op(X),
{\varepsilon}>0,
j\in J
\mid
p(\alpha)
\in V
\subseteq
{\mathfrak{l}}_{1}(\alpha)
\right\},
\\
T(U,\sigma,{\varepsilon},j)
\doteqdot
T^{loc}(U,\sigma{\upharpoonright}U,{\varepsilon},j),
\end{cases}$$ $
(\forall U\in Op(X))
(\forall j\in J)
(\forall{\varepsilon}>0)
(\forall\sigma\in\Gamma(p))
$. Any set $
T^{loc}(U,\sigma{\upharpoonright}U,{\varepsilon},j)
$ for a fixed ${\varepsilon}>0$ is called *${\varepsilon}-$Tube*.

\[15512610\] Notice that $
(\forall U\in Op(X))
(\forall j\in J)
(\forall{\varepsilon}>0)
(\forall\sigma_{U}\in\Gamma_{U}(p))
$ $$T^{loc}(U,\sigma_{U},{\varepsilon},j)
=
\bigcup_{y\in U}
B_{{\mathfrak{E}}_{y},j,{\varepsilon}}(\sigma_{U}(y))$$ where for all $
\gamma
\in
{\mathfrak{E}}_{y}
$ $$B_{{\mathfrak{E}}_{y},j,{\varepsilon}}(\gamma)
\doteqdot
\left\{
\beta\in{\mathfrak{E}}_{y}
\mid
\nu_{j}^{y}\left(\beta-\gamma\right)
<{\varepsilon}\right\}.$$

\[ Neighbourhood’s filter ${\mathcal{I}}_{\alpha}^{\tau}$ \] \[11372310\] Let $
{\mathfrak{P}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},p,X,{\mathfrak{N}}{\right\rangle}}
$ be a bundle of $\Omega-$spaces

1.  if ${\mathfrak{P}}$ is locally full $\forall\alpha\in{\mathfrak{E}}$ and $\forall{\mathfrak{l}}\in{\mathcal{K}}^{loc}$ the class ${\mathcal{B}}_{{\mathfrak{l}}}^{loc}(\alpha)$ is a basis of a filter moreover $${\mathfrak{F}_{{\mathcal{B}}_{{\mathfrak{l}}}^{loc}(\alpha)}^{{\mathfrak{E}}}}
    =
    {\mathcal{I}}_{\alpha}^{\tau};$$

2.  if ${\mathfrak{P}}$ is full or locally full over a completely regular space then $\forall\alpha\in{\mathfrak{E}}$ and $\forall{\mathfrak{l}}\in{\mathcal{K}}$ the class ${\mathcal{B}}_{{\mathfrak{l}}}(\alpha)$ is a basis of a filter moreover $${\mathfrak{F}_{{\mathcal{B}}_{{\mathfrak{l}}}(\alpha)}^{{\mathfrak{E}}}}
    =
    {\mathcal{I}}_{\alpha}^{\tau}.$$

Here ${\mathcal{I}}_{\alpha}^{\tau}$ is the neighbourhood’s filter of $\alpha$ in the topological space ${\left\langle {\mathfrak{E}},\tau{\right\rangle}}$.

Statement $(1)$ follows by applying $1.5.\,VII$ of [@gie], while statement $(2)$ follows by statement $(1)$ and the fact that for all $U\in Op(X)$ and $\sigma\in\Gamma(p)$ we have $
\sigma{\upharpoonright}U
\in\Gamma_{U}(p)
$ and $
T(U,\sigma,{\varepsilon},j)
\doteqdot
T^{loc}(U,\sigma{\upharpoonright}U,{\varepsilon},j),
$ for all $j\in J$ and ${\varepsilon}>0$.

\[12312410\] Let $
{\mathbf{E}}
\doteqdot
\{{\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}\}_{x\in X}
$ be a nice family of Hlcs with $
{\mathfrak{N}}_{x}
\doteqdot
\{
\nu_{j\in J}^{x}
\}
$ for all $x\in X$. Moreover let $
{\mathcal{E}}
$ satisfy $FM(3^{*})-FM(4)$ with respect to ${\mathbf{E}}$. Set as usual $
{\mathfrak{E}}({\mathbf{E}})
\doteqdot
\bigcup_{x\in X}
\{x\}
\times
{\mathbf{E}}_{x}
$ and $$\begin{cases}
{\mathcal{K}}^{{\mathcal{E}}}
\doteqdot
\prod_{(x,v)\in{\mathfrak{E}}}
{\mathcal{K}}_{(x,v)}^{{\mathcal{E}}}
\\
{\mathcal{K}}_{(x,v)}^{{\mathcal{E}}}
\doteqdot
\left\{
(U,f)
\mid
U\in Op(X),
f
\in{\mathcal{E}}
\mid
x\in U,
f(x)=v
\right\}.
\end{cases}$$ Moreover $
\forall(x,v)\in{\mathfrak{E}}({\mathbf{E}})
$ and $\forall{\mathfrak{l}}\in{\mathcal{K}}^{{\mathcal{E}}}$ set $$\label{15022310}
{\mathcal{B}}_{{\mathfrak{l}}}^{{\mathcal{E}}}((x,v))
=
\left\{
T_{{\mathbf{E}}}(V,{\mathfrak{l}}_{2}((x,v)),{\varepsilon},j)
\mid
{\varepsilon}>0,
j\in J,
V\in Op(X)
\mid
x
\in 
V
\subseteq
{\mathfrak{l}}_{1}((x,v))
\right\}.$$

Here $T_{{\mathbf{E}}}(V,{\mathfrak{l}}_{2}((x,v)),{\varepsilon},j)$ has been defined in .

\[ Neighbourhood’s filter ${\mathcal{I}}_{(x,v)}^{\tau({\mathbf{E}},{\mathcal{E}})}$ \] \[14272310\] Assume notations in Definition \[17471910Ba\] and in Definition \[12312410\]. Let $
{\mathbf{E}}
\doteqdot
\{{\left\langle E_{x},{\mathfrak{N}}_{x}{\right\rangle}}\}_{x\in X}
$ be a nice family of Hlcs with $
{\mathfrak{N}}_{x}
\doteqdot
\{
\nu_{j\in J}^{x}
\}
$ for all $x\in X$, moreover let $
{\mathcal{E}}
$ satisfy $FM(3^{*})-FM(4)$ with respect to ${\mathbf{E}}$. Then $
{\mathfrak{V}}({\mathbf{E}},{\mathcal{E}})
\doteqdot
{\left\langle 
{\left\langle {\mathfrak{E}}({\mathbf{E}}),\tau({\mathbf{E}},{\mathcal{E}}){\right\rangle}}
,\pi_{{\mathbf{E}}},X,{\mathfrak{N}}{\right\rangle}}
$ is a full bundle of $\Omega-$spaces and $\forall(x,v)\in{\mathfrak{E}}({\mathbf{E}})$ $${\mathfrak{F}_{{\mathcal{B}}_{{\mathfrak{l}}}^{{\mathcal{E}}}((x,v))}^{{\mathfrak{E}}({\mathbf{E}})}}
=
{\mathcal{I}}_{(x,v)}^{\tau({\mathbf{E}},{\mathcal{E}})}.$$ Here ${\mathcal{I}}_{(x,v)}^{\tau({\mathbf{E}},{\mathcal{E}})}$ is the neighbourhood’s filter of $(x,v)$ in the topological space $
{\left\langle {\mathfrak{E}}({\mathbf{E}}),\tau({\mathbf{E}},{\mathcal{E}}){\right\rangle}}
$.

By Theorem $5.9.$ of [@gie] ${\mathcal{E}}$ and $\Gamma(p_{1})$ are canonically isomorphic as linear spaces, so ${\mathfrak{V}}({\mathbf{E}},{\mathcal{E}})$ is full by \[15012310\]. The statement hence follows by statement $(2)$ of Corollary \[11372310\].

The following corollaries provide conditions under which the topologies over two bundle spaces are equal.

\[15582210\] Let $
{\left\langle {\left\langle {\mathfrak{E}},\tau_{k}{\right\rangle}},p_{k},X,{\mathfrak{N}}_{k}{\right\rangle}}
$ be a full bundle of $\Omega-$spaces or a locally full bundle over a completely regular space $X$, for $k=1,2$. If there exists a map $p$ such that $p=p_{1}=p_{2}$ (equality as maps) and $
\Gamma(p_{1})
=
\Gamma(p_{2})
$ then $
\tau_{1} 
=
\tau_{2}
$.

By statement $(2)$ of Corollary \[11372310\].

\[15592210\] Let us assume the hypothesis and notations in Corollary \[14272310\], moreover let $
{\mathfrak{P}}_{2}
\doteqdot
{\left\langle {\left\langle E,\tau_{2}{\right\rangle}},p_{2},X,{\mathfrak{N}}_{2}{\right\rangle}}$ be a bundle of $\Omega-$spaces and a map $p$ such that $p=\pi_{{\mathbf{E}}}=p_{2}$ as maps. Thus if the following conditions are satisfied

1.  $X$ is compact;

2.  $
    {\mathcal{E}}
    $ and $\Gamma(p_{2})$ are canonically isomorphic as linear spaces

then $\tau({\mathbf{E}},{\mathcal{E}})=\tau_{2}$.

By Theorem $5.9.$ of [@gie] ${\mathcal{E}}$ and $
\Gamma(\pi_{{\mathbf{E}}})
$ are canonically isomorphic as linear spaces if $X$ is compact, so $\Gamma(\pi_{{\mathbf{E}}})=\Gamma(p_{2})$. Moreover \[15012310\] and the shown fact that ${\mathcal{E}}$ and $\Gamma(\pi_{{\mathbf{E}}})$ are canonically isomorphic ensure that ${\mathfrak{V}}({\mathbf{E}},{\mathcal{E}})$ is a full bundle, thus it is so ${\mathfrak{P}}_{2}$ by the equality $\Gamma(\pi_{{\mathbf{E}}})=\Gamma(p_{2})$. Hence the statement follows by Corollary \[15582210\].

### Direct Sum of Bundles of $\Omega-$spaces  {#directsum}

\[[@jar]\] \[18111755\] Let $\{E_{i}\}_{i\in I}$ a family of lcs. Then we denote by $\tau_{0}$, $\tau_{b}$, $\tau_{l}$, $
\tau_{{\mathfrak{l}}}
$ respectively the topology on $\bigoplus_{i\in I}E_{i}$ induced by the product topology on $\prod_{i\in I}E_{i}$, that induced by the box topology on $\prod_{i\in I}E_{i}$ (see [@jar]), the direct sum topology, Ch. $4$, $\S 3$ of [@jar], finally the lc-direct sum topology Ch. $6$, $\S 6$ of [@jar].

\[16322110\] Let ${\left\langle E_{i},\nu_{i}{\right\rangle}}_{i=1}^{n}$ a finite family of lcs where $\nu_{i}=\{\nu_{i,l_{i}}\mid l_{i}\in L_{i}\}$ is a fundamental directed set of seminorms of $E_{i}$. Let us set for all $i=1,...,n$, $l_{i}\in L_{i}$ and $
\rho\in
\prod_{i=1}^{n}
L_{i}
$ $$\begin{cases}
\hat{\nu}_{i,l_{i}}
\doteqdot
\nu_{i,l_{i}}
\circ
\Pr_{i}
\\
\hat{\mu_{\rho}}
\doteqdot
\sum_{i=1}^{n}
\hat{\nu}_{i,l_{i}},
\end{cases}$$ where $
\Pr_{i}:\prod_{k=1}^{n}E_{k}
\ni
x
\mapsto
x_{i}
\in E_{i}
$.

Then $
\hat{\mu}
\doteqdot
\{\hat{\mu}_{\rho}
\mid\rho\in\prod_{i=1}^{n}L_{i}\}
$ is a directed set of seminorms on $
\bigoplus_{i=1}^{n}
E_{i}
$, moreover by setting $$\begin{cases}
{\mathcal{B}}({\mathbf{0}})
\doteqdot
\{
W_{{\varepsilon}}^{\rho}
\mid
{\varepsilon},\rho\in\prod_{i=1}^{n}
L_{i}
\}
\\
W_{{\varepsilon}}^{\rho}
\doteqdot
\{
x\in
\bigoplus_{i=1}^{n}
E_{i}
\mid
\hat{\mu}_{\rho}(x)
<{\varepsilon}\},
\end{cases}$$ we have that $
{\mathcal{B}}({\mathbf{0}})
$ is a base of a filter on $
\bigoplus_{i=1}^{n}
E_{i}
$ in addition $${\mathfrak{F}_{{\mathcal{B}}({\mathbf{0}})}^{\bigoplus_{i=1}^{n}E_{i}}}
=
{\mathcal{I}}_{{\mathbf{0}}}^{\tau},$$ where $\tau$ is the unique locally convex topology on $
\bigoplus_{i=1}^{n}
E_{i}
$ generated by $\hat{\mu}$ and ${\mathcal{I}}_{{\mathbf{0}}}^{\tau}$ is the neighbourhood’s filter of ${\mathbf{0}}$ with respect to the topology $\tau$. Finally with the notations of Definition \[18111755\] we have $
\tau
=
\tau_{0}
=
\tau_{b}
=
\tau_{l}
=
\tau_{{\mathfrak{l}}}
$.

Only in this proof we set $I\doteqdot\{1,...,n\}$, $L\doteqdot\prod_{i\in I}L_{i}$ and $
E^{\oplus}
\doteqdot
\bigoplus_{i=1}^{n}
E_{i}
$. Due to the fact that $n<\infty$ we know that $\prod_{i=1}^{n}E_{i}=E^{\oplus}$ so by [@jar] $\S 4.3.$ the set $
\{\prod_{i=1}^{n}U_{i}\mid U_{i}\in{\mathfrak{(}}U)_{i}\}
$ is a ${\mathbf{0}}-$basis for the box topology on $E^{\oplus}$ if ${\mathfrak{U}}_{i}$ is a ${\mathbf{0}}-$basis for the topology on $E_{i}$. Moreover $\nu_{i}$ is directed so by $II.3$ of [@BourTVS] we can choose $$\label{17442110}
{\mathfrak{U}}_{i}
=
\{
V(\nu_{i,l_{i}},{\varepsilon})
\doteqdot
\{x_{i}\in E_{i}\mid\nu_{i,l_{i}}(x_{i})<{\varepsilon}\}
\mid
{\varepsilon}>0,
l_{i}\in L_{i}
\}.$$ Thus if we set $$\label{18052110}
\begin{cases}
{\mathcal{B}}_{1}({\mathbf{0}})
\doteqdot
\{
U_{\eta}^{\rho}\mid\eta\in({\mathbb {R}}_{0}^{+})^{n},
\rho\in L
\}
\\
U_{\eta}^{\rho}
\doteqdot
\{
x\in E^{\oplus}
\mid
(\forall i\in I)
(\hat{\nu}_{i,\rho_{i}}(x)<\eta_{i})
\},
\end{cases}$$ then ${\mathcal{B}}_{1}({\mathbf{0}})$ is is a ${\mathbf{0}}-$basis for the topology $\tau_{0}$. Moreover $
U_{{\varepsilon}}^{\rho}
=
\bigcap_{i=1}^{n}
V(\hat{\nu}_{i,\rho_{i}}\eta_{i})
$ so if we set $${\mathcal{G}}({\mathbf{0}})
\doteqdot
\left\{
\bigcap_{s\in M}
V(\hat{\nu}_{s}{\varepsilon}_{M}(s))
\mid
M\in
{\mathcal{P}}_{\omega}
\left(\bigcup_{i\in I}\{i\}\times L_{i}\right)
{\varepsilon}_{M}:M\to{\mathbb {R}}_{0}^{+}
\right\},$$ then by $
{\mathcal{B}}_{1}({\mathbf{0}})
\subseteq
{\mathcal{G}}({\mathbf{0}})
$. Moreover by applying $II.3$ of [@BourTVS], ${\mathcal{G}}({\mathbf{0}})$ is a basis of a filter thus $${\mathfrak{F}_{{\mathcal{B}}_{1}({\mathbf{0}})}^{E^{\oplus}}}
\subseteq
{\mathfrak{F}_{{\mathcal{G}}({\mathbf{0}})}^{E^{\oplus}}}.$$ Now for all $M
\in
{\mathcal{P}}_{\omega}
\left(\bigcup_{i\in I}\{i\}\times L_{i}\right)
$ we have $
M=\bigcup_{i\in I}M_{i}
$ with $
M_{i}\doteqdot
M\cap(\{i\}\times L_{i})
=
\{i\}\times Q_{i}
$ for some $Q_{i}\in{\mathcal{P}}_{\omega}(L_{i})$. Hence $
\forall M
\in
{\mathcal{P}}_{\omega}
\left(\bigcup_{i\in I}\{i\}\times L_{i}\right)
$ and $\forall{\varepsilon}_{M}:M\to{\mathbb {R}}_{0}^{+}$ $$\begin{aligned}
T\doteqdot
\bigcap_{s\in M}
V(\hat{\nu}_{s},{\varepsilon}_{M}(s))
&
=
\bigcap_{i\in I}
\bigcap_{s\in M_{i}}
V(\hat{\nu}_{s},{\varepsilon}_{M}(s))
\\
&
=
\bigcap_{i\in I}
\bigcap_{l_{i}\in Q_{i}}
\{
x\in E^{\oplus}
\mid
x_{i}\in V(\nu_{i,l_{i}},{\varepsilon}_{M}(i,l_{i}))
\\
&
=
\bigcap_{i\in I}
\left\{
x\in E^{\oplus}
\mid
x_{i}\in 
\bigcap_{l_{i}\in Q_{i}}
V(\nu_{i,l_{i}},{\varepsilon}_{M}(i,l_{i}))
\right\}.
\end{aligned}$$ Moreover we know that ${\mathfrak{U}}_{i}$ is a basis of a filter on $E_{i}$ thus for aa $i\in I$ there exists $\lambda_{i}>0$ and $k_{i}\in L_{i}$ such that $$V(\nu_{i,k_{i}},\lambda_{i})
\subseteq
\bigcap_{l_{i}\in Q_{i}}
V(\nu_{i,l_{i}},{\varepsilon}_{M}(i,l_{i})),$$ hence $$\begin{aligned}
{\mathcal{G}}({\mathbf{0}})
\ni
T
&
\supseteq
\bigcap_{i\in I}
\{
x\in E^{\oplus}
\mid
x_{i}\in 
V(\nu_{i,k_{i}},\lambda_{i})
\}
\\
&
=
\bigcap_{i\in I}
V(\hat{\nu}_{i,k_{i}},\lambda_{i})
\in
{\mathcal{B}}_{1}({\mathbf{0}}).
\end{aligned}$$ Therefore by a well-known property of filters $
{\mathfrak{F}_{{\mathcal{G}}({\mathbf{0}})}^{E^{\oplus}}}
\subseteq
{\mathfrak{F}_{{\mathcal{B}}_{1}({\mathbf{0}})}^{E^{\oplus}}}
$ then $$\label{18062110}
{\mathfrak{F}_{{\mathcal{G}}({\mathbf{0}})}^{E^{\oplus}}}
=
{\mathfrak{F}_{{\mathcal{B}}_{1}({\mathbf{0}})}^{E^{\oplus}}}.$$ By applying $II.3$ of [@BourTVS] we know that ${\mathfrak{F}_{{\mathcal{G}}({\mathbf{0}})}^{E^{\oplus}}}$ is the ${\mathbf{0}}-$neighbourhood’s filter with respect to the locally convex topology generated by the family of seminorms $
\{\nu_{s}\mid 
s\in
\bigcup_{i\in I}\{i\}\times L_{i}
\}
$ thus by and $$\label{18112110}
\left\{\nu_{s}\mid 
s\in
\bigcup_{i\in I}\{i\}\times L_{i}
\right\}
\text{ is a fss for $\tau_{0}$},$$ where $fss$ is for fundamental system of seminorms.

Now $\hat{\mu}$ is a set of seminorms on $E^{\oplus}$. Let $\rho^{1},\rho^{2}\in L$ then by the hypothesis that $\nu_{i}$ is directed, for all $i\in I$ there exists $\rho_{i}\in L_{i}$ such that $\rho_{i}\geq\rho^{1},\rho^{2}$ thus $\hat{\mu}_{\rho}
\geq
\hat{\mu}_{\rho^{1}},
\hat{\mu}_{\rho^{2}}
$, hence $\hat{\mu}$ is directed. Therefore setting $$\begin{cases}
{\mathcal{B}}({\mathbf{0}})
\doteqdot
\{
W_{{\varepsilon}}^{\rho}
\mid
{\varepsilon}>0,
\rho\in L
\}
\\
W_{{\varepsilon}}^{\rho}
\doteqdot
\{
x\in E^{\oplus}
\mid
\hat{\mu}_{\rho}(x)
<{\varepsilon}\}
\end{cases}$$ by applying $II.3$ of [@BourTVS] $$\label{18242110}
{\mathcal{B}}({\mathbf{0}})
\text{ is the ${\mathbf{0}}-$basis
for the topology
gen. by
$\hat{\mu}$.
}$$ Now $
(\forall(k,l_{k})\in 
\bigcup_{i\in I}\{i\}\times L_{i})
(\exists\,\rho\in L)
(\hat{\nu}_{k,l_{k}}\leq a\hat{\mu}_{\rho})
$ indeed keep any $\rho$ s.t. $\rho(k)=l_{k}$. While $
(\forall\rho\in I)
(m\in{\mathbb{N}})
(\exists\,s_{1},...,s_{m}\in
\bigcup_{i\in I}\{i\}\times L_{i})
(\exists\,a>0)
(\hat{\mu}_{\rho}\leq a
\sup_{r}\hat{\nu}_{s_{r}})
$ indeed it is sufficient ot set $m=n$, $
a=n
$ and $
s_{i}
=
(i,\rho_{i})
$ for all $i\in I$. Therefore by applying Corollary $1$ $II.7$ of [@BourTVS] and by and we have that $\hat{\mu}$ is a fundamental directedset of seminorms for the topology $\tau_{0}$ hence the part of the statement concerning $\tau_{0}$. By Prop. $2$, $\S 3$, Ch $4$ of [@jar] we know that $\tau_{0}
=
\tau_{b}
=
\tau_{l}
$. Finally $
\tau_{{\mathfrak{l}}}
=
\tau_{l}
$ by the fact that $
\tau_{{\mathfrak{l}}}
$ is the finest locally convex topology among those which are coarser than $\tau_{l}$, $\S 6$, Ch $6$ of [@jar], and the just now shown fact that $\tau_{l}$ is locally convex being equal to $\tau_{0}$ which is generated by $\hat{\mu}$.

Now we shall apply the result obtained in the previous proposition, in order to extend to the case of bundles of $\Omega-$spaces what is a standard construction in the Banach bundles case.

\[12422110\] Let $
\{
{\mathfrak{V}}_{i}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}}_{i},\tau_{i}{\right\rangle}},\pi_{i},X,{\mathfrak{N}}_{i}{\right\rangle}}
\}_{i=1}^{n}
$ be a family of bunldes of $\Omega-$spaces, let us denote $
{\mathfrak{N}}_{i}
=
\{
\nu_{i,l_{i}}
\mid
l_{i}\in L_{i}
\}
$ and $
{\mathfrak{N}}_{i}^{x}
=
\{
\nu_{i,l_{i}}^{x}
\doteqdot
\nu_{i,l_{i}}{\upharpoonright}\left({\mathfrak{E}}_{i}\right)_{x}
\mid
l_{i}\in L_{i}
\}
$, with $
\left({\mathfrak{E}}_{i}\right)_{x}
\doteqdot
\overset{-1}{\pi_{i}}(x)
$ for all $i=1,...,n$. Set

1.  $
    {\mathbf{E}}_{x}^{\oplus}
    \doteqdot
    \bigoplus_{i=1}^{n}\left({\mathfrak{E}}_{i}\right)_{x}
    $;

2.  $
    {\mathfrak{n}}_{x}^{\oplus}
    \doteqdot
    \{
    \hat{\mu}_{\rho}^{x}
    \mid
    \rho
    \in
    \prod_{i=1}^{n}
    L_{i}
    \}
    $, where $$\label{18260603}
    \boxed{
    \hat{\mu}_{\rho}^{x}
    =
    \sum_{i=1}^{n}
    \hat{\nu}_{i,\rho_{i}}^{x};
    }$$

3.  $
    {\mathcal{E}}^{\oplus}
    $ is the linear subspace of $
    \prod_{x\in X}
    {\mathbf{E}}_{x}^{\oplus}
    $ generated by the following set $$\label{18270603}
    \bigcup_{i=1}^{n}
    \tilde{\Gamma}(\pi_{i}).$$

Here $
\Pr_{i}^{x}:
{\mathbf{E}}_{x}^{\oplus}
\ni
x
\mapsto
x(i)
\in
\left({\mathfrak{E}}_{i}\right)_{x}
$ while $
\hat{\nu}_{i,\rho_{i}}^{x}
=
\nu_{i,\rho_{i}}^{x}
\circ
\Pr_{i}^{x}
$ and $
I_{i}^{x}:
\left({\mathfrak{E}}_{i}\right)_{x}
\to
{\mathbf{E}}_{x}^{\oplus}
$ is the canonical inclusion, i.e. $
\Pr_{j}^{x}
\circ
I_{i}^{x}
=
\delta_{i,j}
\,
Id^{x}
$, finally $
\tilde{\Gamma}(\pi_{i})
\doteqdot
\{
\tilde{f}
\mid
f\in
\Gamma(\pi_{i})
\}
$, with $
\tilde{f}(x)
\doteqdot
I_{i}^{x}
(f(x))
$.

Notice that $
\{
{\left\langle 
\left({\mathfrak{E}}_{i}\right)_{x},{\mathfrak{N}}_{i}^{x}{\right\rangle}}
\}_{i=1}^{n}
$ for all $x\in X$ is a family of Hlcs where ${\mathfrak{N}}_{i}^{x}$ is a directed family of seminorms defining the topology on $
\left({\mathfrak{E}}_{i}\right)_{x}
$, for all $i=1,...,n$, moreover by Lemma \[19420111\] ${\mathcal{E}}^{\oplus}$ satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{E}}^{\oplus}$. Finally by applying Theorem \[16322110\] we have that ${\mathfrak{n}}_{x}^{\oplus}$ is a *directed* set of seminorms on ${\mathbf{E}}_{x}^{\oplus}$ [^11] so for what before said we can state that the couple $$\label{21142110}
{\left\langle {\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}{\right\rangle}},$$ where $${\mathbf{E}}^{\oplus}
\doteqdot
\left\{
{\left\langle {\mathbf{E}}_{x}^{\oplus},{\mathfrak{n}}_{x}^{\oplus}{\right\rangle}}
\right\}_{x\in X},$$ satisfies the requirements of Proposition $5.8.$ of [@gie]. Therefore generates a bundle of $\Omega-$spaces which according the notations in Definition is \[17471910Ba\] $$\label{14442410}
\bigoplus_{i=1}^{n}
{\mathfrak{V}}_{i}
\doteqdot
{\mathfrak{V}}({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus})$$ and called the *bundle direct sum of the family $
\{
{\mathfrak{V}}_{i}
\}_{i=1}^{n}
$*.

\[22550110\] Note that Theorem \[16322110\] shows much more then the directness of the set of seminorms ${\mathfrak{n}}_{x}^{\oplus}$, indeed it proves that ${\mathfrak{n}}_{x}^{\oplus}$ induces on ${\mathbf{E}}_{x}^{\oplus}$ the product topology.

\[19420111\] ${\mathcal{E}}^{\oplus}$ satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{E}}^{\oplus}$.

$
I_{i}^{x}
$ is a bijective map onto its range whose inverse is $
\Pr_{i}^{x}
{\upharpoonright}Range(I_{i}^{x})
$. Moreover by definition of the product topology $\Pr_{i}^{x}$ is continuous with respect to the topology $\tau_{0}^{i}$ on $Range(I_{i}^{x})$ induced by $\tau_{0}$ [@BourGT Ch.1], while $I_{i}^{x}$ is continuous with respect to $\tau_{0}^{i}$ by [@jar $\S$ 4.3 Pr.1] and the definition of $\tau_{l}$. Hence by Theorem \[16322110\] $
I_{i}^{x}
$ is an isomorphism of the tvs’s $
{\left\langle \left({\mathfrak{E}}_{i}\right)_{x},{\mathfrak{N}}_{i}^{x}{\right\rangle}}
$ and $
I_{i}^{x}
\left(
\left({\mathfrak{E}}_{i}\right)_{x}
\right)
$ as subspace of $
{\left\langle {\mathbf{E}}_{x}^{\oplus},{\mathfrak{n}}_{x}^{\oplus}{\right\rangle}}
$.

Moreover [@gie 1.5.III] and [@gie 1.6.viii] [^12] we deduce that $
\{
\sigma(x)\mid
\sigma\in\Gamma(\pi_{i})
\}
$ is dense in $
{\left\langle \left({\mathfrak{E}}_{i}\right)_{x},{\mathfrak{N}}_{i}^{x}{\right\rangle}}
$. Therefore $\forall i=1,...,n$ and $\forall x\in X$ $$\label{22170111}
\{
I_{i}^{x}(\sigma(x))\mid
\sigma\in\Gamma(\pi_{i})
\}
\text{
is dense
in
$
I_{i}^{x}
\left(
\left({\mathfrak{E}}_{i}\right)_{x}
\right)
$.
}$$ where $
I_{i}^{x}
\left(
\left({\mathfrak{E}}_{i}\right)_{x}
\right)
$ has to be intended as topological vector subspace of $
{\left\langle {\mathbf{E}}_{x}^{\oplus},{\mathfrak{n}}_{x}^{\oplus}{\right\rangle}}
$. So by the continuity of the sum on $
{\left\langle {\mathbf{E}}_{x}^{\oplus},{\mathfrak{n}}_{x}^{\oplus}{\right\rangle}}
$ and the fact that ${\mathbf{E}}_{x}^{\oplus}$ is generated as linear space by the set $
\bigcup_{i=1}^{n}
I_{i}^{x}
\left(
\left(
{\mathfrak{E}}_{i}\right)_{x}\right)
$ we can state $\forall x\in X$ that $$\label{22050111}
\{
F(x)
\mid
F
\in
{\mathcal{E}}^{\oplus}
\}
\text{ 
is dense in 
$
{\left\langle {\mathbf{E}}_{x}^{\oplus},{\mathfrak{n}}_{x}^{\oplus}{\right\rangle}}$.
}$$ Namely by $$(\forall v\in{\mathfrak{E}}^{\oplus})
(\forall i=1,...,n)
(\exists\,
\{
\sigma_{\alpha_{i}}
\}_{\alpha_{i}\in D_{i}}
\text{ net }
\subset
\Gamma(\pi_{i})
)$$ such that $$\begin{aligned}
v
&
=
\sum_{i=1}^{n}I_{i}^{x}(\Pr_{i}^{x}(v))
=
\sum_{i=1}^{n}
\lim_{\alpha_{i}\in D_{i}}
I_{i}^{x}(\sigma_{\alpha_{i}}(x))
\\
&
=
\sum_{i=1}^{n}
\lim_{\alpha\in D}w_{\alpha}^{i}(x)
=
\lim_{\alpha\in D}
\sum_{i=1}^{n}
w_{\alpha}^{i}(x)
\\
&
=
\lim_{\alpha\in D}
\sum_{i=1}^{n}
I_{i}^{x}(\sigma_{\alpha(i)}(x)),
\end{aligned}$$ where $
D
\doteqdot
\prod_{i=1}^{n}
D_{i}
$ while $
w_{\alpha}^{i}(x)
\doteqdot
I_{i}^{x}(\sigma_{\alpha(i)}(x))
$ for all $\alpha\in D$. Moreover $\forall\alpha\in D$ $$\left(
X\ni x\mapsto
\sum_{i=1}^{n}
I_{i}^{x}(\sigma_{\alpha(i)}(x))
\right)
\in
{\mathcal{E}}^{\oplus}$$ then and $FM(3)$ follow.

Finally $FM(4)$ follows by [@gie 1.6.iii] applied to any $\sigma_{i}\in\Gamma(\pi_{i})$ for all $i=1,...,n$ indeed $\forall\sigma_{i}\in\Gamma(\pi_{i})$ $$\hat{
\nu_{i,\rho_{i}}^{x}
}
(\tilde{\sigma}_{i}(x))
=
\nu_{i,\rho_{i}}^{x}
\circ
\Pr_{i}^{x}
\circ
I_{i}^{x}
\circ
\sigma_{i}(x)
=
\nu_{i,\rho_{i}}^{x}
\circ
\sigma_{i}(x).$$

\[21262110\] By Definition \[17471910Ba\] and $$\bigoplus_{i=1}^{n}
{\mathfrak{V}}_{i}
=
{\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}^{\oplus}),\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}){\right\rangle}},\pi_{{\mathbf{E}}^{\oplus}},X,{\mathfrak{n}}^{\oplus}{\right\rangle}}$$ where

1.  $
    {\mathfrak{E}}({\mathbf{E}}^{\oplus})
    \doteqdot
    \bigcup_{x\in X}
    \{x\}
    \times
    {\mathbf{E}}_{x}^{\oplus}
    $, $
    \pi_{{\mathbf{E}}^{\oplus}}:
    {\mathfrak{E}}({\mathbf{E}}^{\oplus})
    \ni
    (x,v)
    \mapsto x
    \in X
    $.

2.  $
    {\mathfrak{n}}^{\oplus}
    =
    \{
    \hat{\mu}_{\rho}:
    \mid
    \rho
    \in
    \prod_{i=1}^{n}
    L_{i}
    \}
    $, with $
    \hat{\mu}_{\rho}:
    {\mathfrak{E}}({\mathbf{E}}^{\oplus})
    \ni
    (x,v)
    \mapsto
    \hat{\mu}_{\rho}^{x}(v)
    $;

3.  $\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus})$ is the topology on ${\mathfrak{E}}({\mathbf{E}}^{\oplus})$ such that for all $
    (x,v)\in
    {\mathfrak{E}}({\mathbf{E}}^{\oplus})
    $ $${\mathcal{I}}_{(x,v)}^{{\mathfrak{E}}({\mathbf{E}}^{\oplus})}
    \doteqdot
    {\mathfrak{F}_{{\mathcal{B}}((x,v))}^{{\mathfrak{E}}({\mathbf{E}}^{\oplus})}}$$ is the neighbourhood’s filter of $(x,v)$ with respect to it. Here ${\mathfrak{F}_{{\mathcal{B}}((x,v))}^{{\mathfrak{E}}({\mathbf{E}}^{\oplus})}}$ is the filter on ${\mathfrak{E}}({\mathbf{E}}^{\oplus})$ generated by the following filter’s base $$\begin{aligned}
    {1}
    {\mathcal{B}}^{\oplus}((x,v))
    \doteqdot
    \{
    T_{{\mathbf{E}}^{\oplus}}(U,\sigma,{\varepsilon},\rho)
    &
    \mid
    U\in Open(X),
    \sigma\in{\mathcal{E}}^{\oplus},
    {\varepsilon}>0,
    \rho
    \in
    \prod_{i=1}^{n}
    L_{i}
    \\
    &
    \mid
    x\in U,
    \hat{\mu}_{\rho}^{x}(v-\sigma(x))
    <{\varepsilon}\},\end{aligned}$$ where $$T_{{\mathbf{E}}^{\oplus}}(U,\sigma,{\varepsilon},\rho)
    \doteqdot
    \left\{
    (y,w)\in{\mathfrak{E}}({\mathbf{E}}^{\oplus})
    \mid
    y\in U,
    \hat{\mu}_{\rho}^{y}(w-\sigma(y))
    <{\varepsilon}\right\};$$

Finally the following is a useful characterization of continuous maps valued in ${\mathfrak{E}}({\mathbf{E}}^{\oplus})$.

\[17571212\] Let $
\{
{\mathfrak{V}}_{i}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}}_{i},\tau_{i}{\right\rangle}},\pi_{i},X,{\mathfrak{N}}_{i}{\right\rangle}}
\}_{i=1}^{n}
$ be a family of bunldes of $\Omega-$spaces, $f\in{\mathfrak{E}}({\mathbf{E}}^{\oplus})^{X}$ and $x\in X$. Then $f$ is continuous in $x$ if and only if $f_{0}^{i}:X\to{\mathfrak{E}}_{i}$ is continuous in $x$ for all $i=1,...,n$, where $
f_{0}
\in
\left(
\bigcup_{z\in X}
{\mathbf{E}}_{z}^{\oplus}
\right)^{X}
$ such that $\forall z\in X$ $
f(z)=(z,f_{0}(z))
$ and $$f_{0}^{i}(z)
\doteqdot
\Pr_{i}^{\pi_{{\mathbf{E}}^{\oplus}}
(f(z))}
\circ
f_{0}(z).$$ In particular $f\in\Gamma(\pi_{{\mathbf{E}}^{\oplus}})$ if and only if $\left(
X\ni z\mapsto
\Pr_{i}^{z}\circ f_{0}(z)
\in({\mathfrak{E}}_{i})_{z}
\right)
\in\Gamma(\pi_{i})$, for all $i=1,...,n$.

By $(1)
\Leftrightarrow
(5)$ in Theorem \[15380512\] applied to $\bigcup_{i=1}^{n}
\tilde{\Gamma}(\pi_{i})$.

\[16392601\] Let $
\{
{\mathfrak{V}}_{i}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}}_{i},\tau_{i}{\right\rangle}},\pi_{i},X,{\mathfrak{N}}_{i}{\right\rangle}}
\}_{i=1}^{n}
$ be a family of bunldes of $\Omega-$spaces, and ${\mathfrak{V}}({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus})$ the bundle direct sum of the family $\{
{\mathfrak{V}}_{i}
\}_{i=1}^{n}$. By construction we have that $\Gamma(\pi_{{\mathbf{E}}^{\oplus}})
\subset
\prod_{x\in X}
\{x\}\times
{\mathbf{E}}_{x}^{\oplus}$. In what follows, except contrary mention, we convein to consider with abuse of language in the obvious manner $$\Gamma(\pi_{{\mathbf{E}}^{\oplus}})
\subset
\prod_{x\in X}
\bigoplus_{i=1}^{n}\left({\mathfrak{E}}_{i}\right)_{x}.$$ Similarly for $\Gamma^{x}(\pi_{{\mathbf{E}}^{\oplus}})$ for any $x\in X$. Moreover in the case in which for any $i=1,...,n$ we have ${\mathfrak{V}}_{i}
={\mathfrak{V}}({\mathbf{E_{i}}},{\mathcal{E_{i}}})$, with obvious meaning of the symbols we consider $$\Gamma(\pi_{{\mathbf{E}}^{\oplus}})
\subset
\prod_{x\in X}
\bigoplus_{i=1}^{n}\left({\mathbf{E}}_{i}\right)_{x}.$$

Main Claim
==========

 Map pre-bundle ${\mathcal{M}}$ relative to a map system 
---------------------------------------------------------

\[Map Systems\] \[17471910\] ${\left\langle X,{\mathbf{E}},{\mathcal{S}}{\right\rangle}}$ is a *Map System* if

1.  $X$ is a set;

2.  $
    {\mathbf{E}}
    =
    \{
    {\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}
    \}_{x\in X}
    $ is a nice family of Hlcs with $
    {\mathfrak{N}}_{x}
    \doteqdot
    \{\nu^{x}_{j}\mid j\in J\}
    $ for all $x\in X$;

3.  $
    (\exists L\ne\emptyset)
    ({\mathcal{S}}
    =
    \{S_{x}\}_{x\in X})
    $ where $
    S_{x}
    \doteqdot
    \{
    B_{l}^{x}
    \mid
    l\in L
    \}
    \subseteq
    Bounded({\mathbf{E}}_{x})
    $ and $
    \bigcup_{l\in L}
    B_{l}^{x}
    $ is total in ${\mathbf{E}}_{x}$ for all $x\in X$.

\[ Map pre-bundle \] \[17471910A\] We say that $
{\mathbf{M}}
$ is a *Map pre-bundle relative to ${\left\langle X,Y,{\mathbf{E}},{\mathcal{S}}{\right\rangle}}$* if

1.  ${\left\langle X,{\mathbf{E}},{\mathcal{S}}{\right\rangle}}$ is a map system;

2.  $
    {\mathbf{M}}
    =
    \{
    {\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}}
    \}_{x\in X}
    $ is a nice family of Hlcs;

3.  $Y$ is a Hausdorff topological space and $\forall x\in X$ $$\boxed{
    \begin{aligned}
    {\mathbf{M}}_{x}
    &
    \subseteq
    {\mathcal{C}_{c}
    \left(Y,{\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})\right)};
    \\
    {\mathfrak{R}}_{x}
    &
    =
    \left\{
    \sup_{(K,j,l)\in{\mathcal{O}}}
    q_{(K,j,l)}^{x}
    {\upharpoonright}{\mathbf{M}}_{x}
    \mid
    {\mathcal{O}}\in{\mathcal{P}}_{\omega}\left(Comp(Y)
    \times J\times L\right)
    \right\}.
    \end{aligned}
    }$$

Here we recall that ${\mathcal{P}}_{\omega}(A)$ is the class of all finite parts of the set $A$, ${\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})$, for all $x\in X$, is the $lcs$ of all continuous linear maps ${\mathcal{L}}({\mathbf{E}}_{x})$ on ${\mathbf{E}}_{x}$ with the topology of uniform convergence over the sets in $S_{x}$, hence its topology is generated by the following set of seminorms $$\label{21532606}
\left\{
p_{j,l}^{x}:
{\mathcal{L}}({\mathbf{E}}_{x})
\ni
\phi
\mapsto
\sup_{v\in B_{l}^{x}}
\nu_{j}^{x}(\phi(v))
\mid
l\in L,
j\in J
\right\}.$$ Thus by the totality hypothesis and by [@BourTVS Prop. $3$, $III.15$] ${\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})$ is Hausdorff. Finally for all $(K,j,l)
\in
Comp(Y)
\times
J
\times
L$ we set $$\label{21552606}
q_{(K,j,l)}^{x}:
{\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})\right)}
\ni
f
\mapsto
\sup_{t\in K}
p_{j,l}^{x}
(f(t))$$

By the fact that $\{t\}$ is compact for all $t\in Y$ we have that $\bigcup_{K\in Comp(Y)}K=Y$ thus by the shown fact that ${\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})$ is Hausdorff we deduce by [@BourGT Prp. $(1)$, $\S 1.2$, Ch $10$] that ${\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})\right)}$ is Hausdorff. Moreover by [@BourGT Def. $(1)$, $\S 1.1$, Ch $10$] and by the fact that is a $f.s.s.$ on ${\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})$, we can deduce that $\left\{
\sup_{(K,j,l)\in{\mathcal{O}}}
q_{(K,j,l)}^{x}
\mid
{\mathcal{O}}
\in{\mathcal{P}}_{\omega}
\left(
Comp(Y)
\times
J
\times
L
\right)
\right\}$ is a *direct* $f.s.s.$ on ${\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})\right)}$. Hence ${\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}}$ is a topological vector subspace of ${\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})\right)}$ so it is Hausdorff, hence by the construction of ${\mathfrak{R}}_{x}$ we can state that $\{
{\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}}
\}_{x\in X}$ is a nice family of Hlcs in agreement with request $(2)$ in Def. \[17471910A\].

For the sake of completeness and the large use of it in all the work, we shall use Definition \[17471910Ba\] for giving in the following remark the explicit form of ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$.

\[17471910B\] Let $
{\mathbf{M}}
=
\{
{\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}}
\}_{x\in X}
$ be a map pre-bundle relative to $
{\left\langle X,Y,{\mathbf{E}}={\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}_{x\in X},
{\mathcal{S}}{\right\rangle}}
$, moreover let ${\mathcal{M}}$ satisfy $FM(3)-FM(4)$ with respect to $
{\mathbf{M}}
$. Let us denote $
{\mathfrak{N}}_{x}
=
\{
\nu_{j}^{x}
\mid
j\in J
\}
$ for all $x\in X$ and use the notations in Definition \[17471910A\]. Thus for the bundle ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ generated by the couple ${\left\langle {\mathbf{M}},{\mathcal{M}}{\right\rangle}}$ we have

1.  $
    {\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})
    =
    {\left\langle {\left\langle {\mathfrak{E}}({\mathbf{M}}),\tau({\mathbf{M}},{\mathcal{M}}){\right\rangle}}
    ,\pi_{{\mathbf{M}}},X,{\mathfrak{R}}{\right\rangle}}
    $;

2.  $
    {\mathfrak{E}}({\mathbf{M}})
    \doteqdot
    \bigcup_{x\in X}
    \{x\}
    \times
    {\mathbf{M}}_{x}
    $, $
    \pi_{{\mathbf{M}}}
    :{\mathfrak{E}}({\mathbf{M}})
    \ni
    (x,f)
    \mapsto x
    \in X
    $;

3.  $
    {\mathfrak{R}}
    =
    \left\{
    \sup_{(K,j,l)\in{\mathcal{O}}}
    q_{(K,j,l)}
    \mid
    {\mathcal{O}}
    \in{\mathcal{P}}_{\omega}
    \left(
    Comp(Y)
    \times
    J
    \times
    L
    \right)
    \right\}$, with $
    q_{(K,j,l)}:
    {\mathfrak{E}}({\mathbf{M}})
    \ni
    (x,f)
    \mapsto
    q_{(K,j,l)}^{x}(f)$;

4.  $\tau({\mathbf{M}},{\mathcal{M}})$ is the topology on $
    {\mathfrak{E}}({\mathbf{M}})
    $ such that for all $(x,f)\in{\mathfrak{E}}({\mathbf{M}})$ $${\mathcal{I}}_{(x,f)}^{{\mathfrak{E}}({\mathbf{M}})}
    \doteqdot
    {\mathfrak{F}_{{\mathcal{B}}_{{\mathbf{M}}}((x,f))}^{{\mathfrak{E}}({\mathbf{M}})}}$$ is the neighbourhood’s filter of $(x,f)$ with respect to it. Here ${\mathfrak{F}_{{\mathcal{B}}_{{\mathbf{M}}}((x,f))}^{{\mathfrak{E}}({\mathbf{M}})}}$ is the filter on ${\mathfrak{E}}({\mathbf{M}})$ generated by the following filter’s base $$\begin{aligned}
    {1}
    &
    {\mathcal{B}}_{{\mathbf{M}}}((x,f))
    \doteqdot
    \{
    T_{{\mathbf{M}}}
    \left(
    U,\sigma,{\varepsilon},
    {\mathcal{O}}
    \right)
    \mid
    U\in Open(X),
    \sigma\in{\mathcal{M}},
    {\varepsilon}>0,\\
    &
    {\mathcal{O}}
    \in{\mathcal{P}}_{\omega}
    \left(
    Comp(Y)
    \times
    J
    \times
    L
    \right)
    \mid
    x\in U,
    \sup_{(K,j,l)\in{\mathcal{O}}}
    q_{(K,j,l)}^{x}
    (f-\sigma(x))
    <{\varepsilon}\},\end{aligned}$$ where $
    \forall
    U\in Open(X),
    \sigma\in{\mathcal{M}},
    {\varepsilon}>0
    $ and $\forall{\mathcal{O}}
    \in{\mathcal{P}}_{\omega}
    \left(
    Comp(Y)
    \times
    J
    \times
    L
    \right)$ $$T_{{\mathbf{M}}}
    \left(
    U,\sigma,{\varepsilon},
    {\mathcal{O}}
    \right)
    \doteqdot
    \left\{
    (y,g)\in{\mathfrak{E}}({\mathbf{M}})
    \mid
    y\in U,
    \sup_{(K,j,l)\in{\mathcal{O}}}
    q_{(K,j,l)}^{y}
    (g-\sigma(y))
    <{\varepsilon}\right\}.$$

In Remark \[17471910B\] we gave explicitly the neighbourhood’ filters for the topology $\tau({\mathbf{M}},{\mathcal{M}})$. A simpler form for this filter can be obtained with additional requirement as we showed in Corollary \[14272310\].

\[18070512\] Let $
{\mathbf{M}}
=
\{
{\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}}
\}_{x\in X}
$ be a map pre-bundle relative to $
{\left\langle X,Y,{\mathbf{E}}={\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}_{x\in X},
{\mathcal{S}}{\right\rangle}}
$, moreover let ${\mathcal{M}}$ satisfy $FM(3)-FM(4)$ with respect to $
{\mathbf{M}}
$. Thus by Remark \[15412610\] $
\forall
U\in Open(X),
\sigma\in{\mathcal{M}},
{\varepsilon}>0
$ and $\forall
{\mathcal{O}}\in{\mathcal{P}}_{\omega}\left(Comp(Y)
\times J\times L\right)$ $$T_{{\mathbf{M}}}
(U,\sigma,{\varepsilon},
{\mathcal{O}})
=
\bigcup_{y\in U}
B_{{\mathbf{M}}_{y},
{\mathcal{O}},{\varepsilon}}(\sigma(y))$$ where for all $
s
\in
{\mathbf{M}}_{y}
$ $$B_{{\mathbf{M}}_{y},
{\mathcal{O}},{\varepsilon}}(s)
\doteqdot
\left\{
(y,f)\in{\mathfrak{E}}({\mathbf{M}})_{y}
\mid
\sup_{(K,j,l)\in{\mathcal{O}}}
q_{(K,j,l)}^{y}
\left(f-s\right)
<{\varepsilon}\right\}.$$

By applying Remark \[17150312\] we have the following

\[16422010\] Let ${\mathbf{M}}$ be a map pre-bundle relative to $
{\left\langle X,Y,{\mathbf{E}},{\mathcal{S}}{\right\rangle}}
$, moreover let ${\mathcal{M}}$ satisfy $FM(3)-FM(4)$ with respect to ${\mathbf{M}}$. Thus

1.  ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ is a bundle of $\Omega-$spaces;

2.  with the notations of Definition \[17471910B\] ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ is such that

    1.  ${\left\langle {\mathfrak{E}}({\mathbf{M}})_{x},\tau({\mathbf{M}},{\mathcal{M}}){\right\rangle}}$ as topological vector space is isomorphic to ${\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}}$ for all $x\in X$;

    2.  ${\mathcal{M}}$ is canonically isomorphic with a linear subspace of $\Gamma(\pi_{{\mathbf{M}}})$ and if $X$ is compact $
        {\mathcal{M}}
        \simeq
        \Gamma(\pi_{{\mathbf{M}}})
        $.

 $\left(\Theta,{\mathcal{E}}\right)-$structures General case  {#01121953}
-------------------------------------------------------------

In Definition \[10282712\] we show how to generalize the topology of uniform convergence to the case of a bundle ${\left\langle {\mathfrak{M}},\rho,X{\right\rangle}}$ of $\Omega-$spaces, such that $\{{\mathfrak{M}}_{x}\}_{x\in X}$ is a map pre-bundle relative to ${\left\langle X,Y,\{{\mathfrak{E}}_{x}\}_{x\in X},{\mathcal{S}}{\right\rangle}}$, where ${\left\langle {\mathfrak{E}},\pi,X{\right\rangle}}$ is a bundle ${\left\langle {\mathfrak{M}},\rho,X{\right\rangle}}$ of $\Omega-$spaces. The aim is to correlate the topology on ${\mathfrak{M}}$ with that on ${\mathfrak{E}}$ in order to generalize the correlation established in the introduction for the trivial bundle case.

\[15531102\] $$(\bullet):
\prod_{x\in X}({\mathfrak{E}}_{x})^{{\mathfrak{E}}_{x}}
\times
\prod_{x\in X}{\mathfrak{E}}_{x}
\to
\prod_{x\in X}{\mathfrak{E}}_{x}$$ such that for all $F\in
\prod_{x\in X}({\mathfrak{E}}_{x})^{{\mathfrak{E}}_{x}},
v\in\prod_{x\in X}{\mathfrak{E}}_{x}$ we have $$(F\bullet w)(x)
\doteqdot
F(x)(w(x)).$$

\[ **$\left(\Theta,{\mathcal{E}}\right)-$structures**\] \[10282712\] We say that ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ is a *$\left(\Theta,{\mathcal{E}}\right)-$structure* if

1.  $
    {\mathfrak{V}}
    \doteqdot
    {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}
    $ is a bundle of $\Omega-$spaces;

2.  ${\mathcal{E}}\subseteq\Gamma(\pi)$;

3.  $\Theta\subseteq
    \prod_{x\in X}
    Bounded({\mathfrak{E}}_{x})$;

4.  $\forall B\in\Theta$

    1.  ${\mathbf{D}}(B,{\mathcal{E}})\ne\emptyset$;

    2.  $\bigcup_{B\in\Theta}{\mathcal{B}}_{B}^{x}$ is total in ${\mathfrak{E}}_{x}$ for all $x\in X$;

5.  ${\mathfrak{W}}
    \doteqdot
    {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ is a bundle of $\Omega-$spaces such that $\{{\left\langle {\mathfrak{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}}\}_{x\in X}$ is a map pre-bundle relative to ${\left\langle X,Y,\{{\left\langle {\mathfrak{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}
    \}_{x\in X},{\mathcal{S}}{\right\rangle}}$.

Here ${\mathcal{S}}
\doteqdot
\{S_{x}\}_{x\in X}$ and $(\forall B\in\Theta)(\forall x\in X)$ $$\label{11232712}
\boxed{
\begin{cases}
{\mathbf{D}}(B,{\mathcal{E}})
\doteqdot
{\mathcal{E}}
\cap
\left(\prod_{x\in X}B_{x}\right)
\\
{\mathcal{B}}_{B}^{x}
\doteqdot
\{v(x)\mid v\in{\mathbf{D}}(B,{\mathcal{E}})\}
\}
\\
S_{x}
\doteqdot
\{{\mathcal{B}}_{B}^{x}
\mid B\in\Theta\}.
\end{cases}
}$$ Moreover ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ is an *invariant $\left(\Theta,{\mathcal{E}}\right)-$ structure* if it is a $\left(\Theta,{\mathcal{E}}\right)-$ structure such that $$\label{18112502pre}
\left\{
F\in\prod_{z\in X}^{b}
{\mathfrak{M}}_{z}
\mid
(\forall t\in Y)
(F_{t}
\bullet
{\mathcal{E}}(\Theta)
\subseteq
\Gamma(\pi))
\right\}
=
\Gamma(\rho).$$ Finally ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ is a *compatible $\left(\Theta,{\mathcal{E}}\right)-$structure* if it is a $\left(\Theta,{\mathcal{E}}\right)-$structure such that for all $t\in Y$ $$\label{18011202}
\Gamma(\rho)_{t}
\bullet
{\mathcal{E}}(\Theta)
\subseteq
\Gamma(\pi).$$ Here $${\mathcal{E}}(\Theta)
\doteqdot
\bigcup_{B\in\Theta}
{\mathbf{D}}(B,{\mathcal{E}}),$$ and $S_{t}
\doteqdot
\{
F_{t}\mid
F\in S
\}$ and $F_{t}
\in\prod_{x\in X}
{\mathcal{L}}({\mathfrak{E}}_{x})$ such that $F_{t}(x)\doteqdot F(x)(t)$, for all $S\subseteq\prod_{x\in X}
{\mathcal{L}}({\mathfrak{E}}_{x})^{Y}$ $t\in Y$, and $F\in S$.

\[22442606\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure. Then for all $x\in X$ $$\label{15073006}
{\mathfrak{R}}_{x}=
\left\{
\sup_{(K,j,B)\in O}
q_{(K,j,B)}^{x}{\upharpoonright}{\mathfrak{M}}_{x}
\mid
O\in{\mathcal{P}}_{\omega}
\left(Comp(Y)\times J\times \Theta\right)
\right\}$$ where by using the notations of Def. \[10282712\] we set ${\mathfrak{N}}=\{\nu_{j}^{x}\mid j\in J\}$ and for all $K\in Comp(Y)$,$
j\in J$, $B\in\Theta$ $$\label{22542906}
\boxed{
q_{(K,j,B)}^{x}:
{\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)}
\ni f_{x}
\mapsto
\sup_{t\in K}\sup_{v\in{\mathbf{D}}(B,{\mathcal{E}})}
\nu_{j}^{x}\left(f_{x}(t)v(x)\right)
}$$

\[21022406\] Let ${\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ be a bundle, $
{\mathbf{M}}
=
\{
{\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}}
\}_{x\in X}
$ a map pre-bundle relative to ${\left\langle X,Y,\{{\left\langle {\mathfrak{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}
\}_{x\in X},{\mathcal{S}}{\right\rangle}}$ and ${\mathcal{M}}$ satisfy $FM(3)-FM(4)$ with respect to $
{\mathbf{M}}
$. Then Rmk. \[17471910B\] allows us to construct ${\mathfrak{W}}$ satisfying the condition $(5)$ in Def. \[10282712\]. About the problem of satisfying $FM(3)-FM(4)$ recall the very important Rmk. \[21132406\].

The following characterization of ${\mathcal{U}}\in\Gamma_{U}^{x_{\infty}}(\rho)$ will be very important in the sequel.

\[15482712\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure, $x_{\infty}\in W\subseteq X$ and ${\mathcal{U}}\in\prod_{x\in W}^{b}{\mathfrak{M}}_{x}$. By using the notations in Definition \[10282712\] we have $
(1)
\Leftarrow
(2)
\Leftarrow
(3)
\Leftrightarrow
(4)
$ moreover if ${\mathfrak{W}}$ is locally full $(1)
\Leftrightarrow
(2)
\Leftrightarrow
(3)
\Leftrightarrow
(4)
$, finally if ${\mathfrak{W}}$ is full we can choose $U=X$ in $(2)$ and $U'=X$ in $(3)$ and $(4)$. Here

1.  ${\mathcal{U}}\in\Gamma_{W}^{x_{\infty}}(\rho)$;

2.  $
    (\exists\,U\in Op(X)\mid U\ni x_{\infty})
    (\exists\,F\in\Gamma_{U}(\rho))
    (F(x_{\infty})
    ={\mathcal{U}}(x_{\infty}))
    $ such that $
    (\forall j\in J)
    (\forall K\in Compact(Y))
    (\forall B\in\Theta)
    $ $$\label{02022912}
    \boxed{
    \lim_{z\to x_{\infty},z\in W\cap U}
    \sup_{t\in K}
    \sup_{v\in{\mathbf{D}}(B,{\mathcal{E}})}
    \nu_{j}\left({\mathcal{U}}(z)(t)v(z)-F(z)(t)v(z)
    \right)=0;}$$

3.  $
    (\exists\,U'\in Op(X)\mid U'\ni x_{\infty})
    (\exists\,F\in\Gamma_{U'}(\rho))
    (F(x_{\infty})
    ={\mathcal{U}}(x_{\infty}))
    $ and $
    (\forall U\in Op(X)\mid U\ni x_{\infty})
    (\forall\,F\in\Gamma_{U}(\rho)
    \mid
    F(x_{\infty})
    ={\mathcal{U}}(x_{\infty}))
    $ we have $(\forall j\in J)
    (\forall K\in Compact(Y))
    (\forall B\in\Theta)$;

4.  $
    (\exists\,U'\in Op(X)\mid U'\ni x_{\infty})
    (\exists\,F\in\Gamma_{U'}(\rho))
    (F(x_{\infty})
    ={\mathcal{U}}(x_{\infty}))
    $ and ${\mathcal{U}}\in\Gamma_{W}^{x_{\infty}}(\rho)$.

Apply Corollary \[28111707\] to Definition \[17471910A\].

\[11121102\] Let us assume the hypotheses of Lemma \[15482712\] and that ${\mathfrak{W}}$ is full. Moreover let $B\in\Theta$ and $v\in
{\mathbf{D}}(B,{\mathcal{E}})$. Then $(1)
\Rightarrow
(2)$, where

1.  ${\mathcal{U}}\in\Gamma_{W}^{x_{\infty}}(\rho)$ and $\exists\,F\in\Gamma(\rho)$ such that $F(x_{\infty})
    ={\mathcal{U}}(x_{\infty})$ and $(\forall t\in Y)
    (F(\cdot)(t)
    \bullet 
    v\in\Gamma(\pi))$;

2.  $(\forall t\in X)
    ({\mathcal{U}}(\cdot)(t)\bullet v\in
    \Gamma_{W}^{\infty}(\pi))$.

By the position $(1)$ and by the implication $(1)
\Rightarrow
(3)$ of Lemma \[15482712\] and by the fact that the union of all compact subsets of $Y$ is $Y$, being locally compact, we deduce that $(\exists\,F\in\Gamma(\rho))
(F(x_{\infty})
={\mathcal{U}}(x_{\infty}))$ such that $(\forall j\in J)(\forall t\in Y)
(\forall B\in\Theta)$ and $\forall v\in{\mathbf{D}}(B,{\mathcal{E}})$ $$\begin{cases}
\lim_{z\to x_{\infty},z\in W}
\nu_{j}
\left(
{\mathcal{U}}(z)(t)v(z)-F(z)(t)v(z)
\right)
=0,
\\
F(\cdot)(t)\bullet v
\in\Gamma(\pi).
\end{cases}$$ Thus the statement follows by implication $(3)\Rightarrow(1)$ of Corollary \[28111707\].

In general condition $(1)$ is much more stronger than $(2)$. Let us conclude this section with two results constructing a $\left(\Theta,{\mathcal{E}}\right)-$structure and describing $\Gamma^{x_{\infty}}(\rho)$ when ${\mathfrak{V}}$ is trivial.

\[22312406\] Let $Z$ be a normed space $X,Y$ be two topological spaces. Set for all $x\in X$ and $v\in{\mathcal{C}_{b}
\left(X,Z\right)}$ $$\begin{cases}
\begin{aligned}
{\mathcal{M}}
\doteqdot
&
\{
F\in{\mathcal{C}_{b}
\left(X,{\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{s}(Z)\right)}\right)}
\mid
(\forall K\in Comp(Y))
\\
&
(C(F,K)\doteqdot
\sup_{(x,s)\in X\times K}
\|F(x)(s)\|_{B(Z)}<\infty)
\},
\end{aligned}
\\
{\mathbf{M}}_{x}
\doteqdot{\overline}{\{
F(x)\mid F\in{\mathcal{M}}
\}},
\\
\mu_{(v,x)}^{K}:
{\mathbf{M}}_{x}\ni G\mapsto
\sup_{s\in K}
\|G(s)v(x)\|,
\\
{\mathcal{A}}_{x}
\doteqdot\{\mu_{(w,x)}^{K}
\mid
K\in Comp(Y),
w\in{\mathcal{C}_{b}
\left(X,Z\right)}
\},
\\
{\mathbf{M}}
\doteqdot\{
{\left\langle {\mathbf{M}}_{x},{\mathcal{A}}_{x}{\right\rangle}}\}_{x\in X}.
\end{cases}$$ closure in ${\mathcal{C}_{c}
\left(Y,B_{s}(Z)\right)}$. Then ${\mathcal{M}}$ satisfies $FM3-FM4$ with respect to ${\mathbf{M}}$

$FM(3)$ is true by construction, let $v\in{\mathcal{C}_{b}
\left(X,Z\right)}$, $K\in Comp(Y)$, $F\in{\mathcal{M}}$, then $$\sup_{x\in X}
\mu_{(v,x)}^{K}(F(x))
\leq
\sup_{(x,s)\in X\times K}
\|F(x)(s)\|_{B(Z)}
\sup_{x\in X}
\|v(x)\|
<\infty.$$ For all $x,x_{0}\in X$ $$\label{15472506}
\mu_{(v,x)}^{K}(F(x))
\leq
C
\|v(x)-v(x_{0})\|
+
\sup_{s\in K}
\|F(x)(s)v(x_{0})\|,$$ where $C
\doteqdot
\sup_{(x,s)\in X\times K}
\|F(x)(s)\|_{B(Z)}$. Moreover the map ${\mathcal{C}_{c}
\left(Y,B_{s}(Z)\right)}
\ni
f
\mapsto
\sup_{s\in K}\|f(s)w\|
\in
{\mathbb {R}}^{+}$, for all $w\in Z$ is a continuous seminorm, hence by the continuity of $F$ also the map $X\ni x\mapsto 
\sup_{s\in K}\|F(x)(s)w\|
\in
{\mathbb {R}}^{+}$ is continuous. So by $\varlimsup_{x\to x_{0}}
\mu_{(v,x)}^{K}(F(x))
\leq
\sup_{s\in K}
\|F(x_{0})(s)v(x_{0})\|
=
\mu_{(v,x_{0})}^{K}(F(x_{0}))$, and by [@BourGT $(15)$, $\S 5.6$] we have $$\varlimsup_{x\to x_{0}}
\mu_{(v,x)}^{K}(F(x))
=
\mu_{(v,x_{0})}^{K}(F(x_{0})).$$ Therefore by [@BourGT $(13)$, $\S 5.6$], [@BourGT Prp. $3$, $\S 6.2$], and the fact that any map $g$ is $u.s.c.$ at a point iff $-g$ is $l.s.c.$, we can state that $X\ni x\mapsto \mu_{(v,x)}^{K}(F(x))$ is $u.s.c.$ at $x_{0}$ for all $x_{0}\in X$, hence it is $u.s.c.$, which is the $FM(4)$ condition.

By Lemma \[22312406\] and Def. \[17471910Ba\] we can construct the bundle ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ generated by the couple ${\left\langle {\mathbf{M}},{\mathcal{M}}{\right\rangle}}$. In the following result I will construct a $\left(\Theta,{\mathcal{E}}\right)-$structure and describe a large subclass of $\Gamma^{x_{\infty}}(\rho)$.

\[22372406\] Let us assume the notations and hypotheses of Lemma \[22312406\], let ${\mathfrak{V}}$ be the trivial Banach bundle with constant stalk $Z$ and set $\Theta
\doteqdot
\left\{
B_{v}
\mid
v\in{\mathcal{C}_{b}
\left(X,Z\right)}
\right\}$. Then

1.  ${\left\langle {\mathfrak{V}},{\mathfrak{V}}({\mathbf{M}},{\mathcal{M}}),X,Y{\right\rangle}}$ is a $(\Theta,{\mathcal{C}_{b}
    \left(X,Z\right)})-$ structure, moreover if $X$ is compact and $Y$ is locally compact then it is compatible;

2.  Let $f\in\prod_{x\in\ X}{\mathbf{M}}_{x}$ be such that $\sup_{(x,s)\in X\times K}
    \|f(x)(s)\|_{B(Z)}<\infty$ for all $K\in Comp(Y)$ then $
    (a)
    \Leftrightarrow
    (b)
    \Leftrightarrow
    (c)
    \Leftrightarrow
    (d)$, where

    1.  $f\in\Gamma^{x_{\infty}}(\pi_{{\mathbf{M}}})$;

    2.  $(\forall K\in Comp(Y))
        (\forall v\in{\mathcal{C}_{b}
        \left(X,Z\right)})$ $$\lim_{x\to x_{\infty}}
        \sup_{s\in K}
        \|
        f(x)(s)v(x)
        -
        f(x_{\infty})(s)v(x)
        \|
        =0$$

    3.  $f:X\to{\mathcal{C}_{c}
        \left(Y,B_{s}(Z)\right)}$ continuous at $x_{\infty}$;

    4.  $(\forall K\in Comp(Y))
        (\forall w\in Z)$ $$\lim_{x\to x_{\infty}}
        \sup_{s\in K}
        \|
        f(x)(s)w
        -
        f(x_{\infty})(s)w
        \|
        =0.$$

By Rmk. \[21022406\] and Lemma \[22312406\] we have that $(5)$ of Def. \[10282712\] follows. $\Gamma(\pi)\simeq{\mathcal{C}_{b}
\left(X,Z\right)}$ hence by Rmk. \[21500412b\] the others requests of Def. \[10282712\] follow. Thus the first sentence of statement $(1)$. If $X$ is compact by Lemma \[22312406\] and Rmk. \[17150312\] follows that ${\mathcal{M}}\simeq\Gamma(\pi_{{\mathbf{M}}})$, moreover by Rmk. \[21500412b\] we have ${\mathcal{E}}(\Theta)={\mathcal{E}}$ and finally ${\mathcal{E}}\doteq
\Gamma(\pi)\simeq{\mathcal{C}_{b}
\left(X,Z\right)}
$. Hence the second sentence of statement $(1)$ follows if we show that ${\mathcal{M}}_{t}
\bullet
{\mathcal{C}_{b}
\left(X,Z\right)}
\subseteq
{\mathcal{C}_{b}
\left(X,Z\right)}$. To this end fix $v\in{\mathcal{C}_{b}
\left(X,Z\right)}$, $F\in{\mathcal{M}}$, $s\in Y$ and $K_{s}$ a compact neighbourhood of $s$, which there exists by the hypothesis that $Y$ is locally compact. Then we have for all $x,x_{0}\in X$ $$\begin{aligned}
\label{16102506}
&
\|
F(x)(s)v(x)-F(x_{\infty})(s)v(x_{0})
\|
\leq
\\
&
C(F,K_{s})
\|v(x)-v(x_{0})\|
+
\|\left(F(x)(s)-
F(x_{0})(s)
\right)v(x_{0})\|
\end{aligned}$$ By considering that $F\in{\mathcal{C}_{b}
\left(X,{\mathcal{C}_{c}
\left(Y,B_{s}(Z)\right)}\right)}$ and that $s\in K_{s}$ we have that $\lim_{x\to x_{0}}
\|(F(x)(s)-F(x_{0})(s))v(x_{0})\|
=0$. Hence by we deduce that $F_{s}\bullet v$ is continuous at $x_{0}$, so continuous on $X$, in particular $X$ being compact it is also $\|\cdot\|_{Z}-$bounded. Thus $F_{s}\bullet v\in{\mathcal{C}_{b}
\left(X,Z\right)}$ and the second sentence of the statement follows.

Fix $f\in\prod_{x\in\ X}{\mathbf{M}}_{x}$ such that $(\forall K\in Comp(Y))
(\sup_{(x,s)\in X\times K}
\|f(x)(s)\|_{B(Z)}<\infty)$. $(a)\Leftrightarrow(b)$ follows by Lemma \[15482712\], the fact that ${\mathcal{M}}\subseteq\Gamma(\pi_{{\mathbf{M}}})$ by Rmk. \[17150312\], and by $(H:X\ni x\mapsto f(x_{\infty})\in
{\mathcal{C}_{c}
\left(Y,B_{s}(Z)\right)})\in{\mathcal{M}}$, indeed $H$ it is bounded and continuous being constant, moreover $\sup_{(x,s)\in X\times K}
\|H(x)(s)\|_{B(Z)}
=
\sup_{s\in K}
\|f(x_{\infty})(s)\|_{B(Z)}
<\infty$, for all $K\in Comp(Y)$. $(b)\Rightarrow(d)$ follows by the fact that $(X\ni x\mapsto w\in Z)\in{\mathcal{C}_{b}
\left(X,Z\right)}$, and $(c)\Leftrightarrow(d)$ is trivial. For all $K\in Comp(Y)$, $x\in X$ and $s\in K$ $$\begin{aligned}
\|
(f(x)(s)
-
f(x_{\infty})(s))
v(x)
\|
&
\leq
\\
\|
f(x)(s)v(x)
-
f(x_{\infty})(s)v(x_{\infty})
\|
+
\|
f(x_{\infty})(s)v(x_{\infty})
-
f(x_{\infty})(s)v(x)
\|
&
\leq
\\
\|f(x)(s)(v(x)-v(x_{\infty}))\|
+
\|(f(x)(s)-f(x_{\infty})(s))v(x_{\infty})\|
+
\|f(x_{\infty})(s)(v(x_{\infty})-v(x))\|
&
\leq
\\
\left(
\|f(x)(s)\|
+
\|f(x_{\infty})(s)\|
\right)
\|v(x_{\infty})-v(x)\|
+
\|(f(x)(s)-f(x_{\infty})(s))v(x_{\infty})\|
&
\leq
\\
C(f,K)\|v(x_{\infty})-v(x)\|
+
\|(f(x)(s)-f(x_{\infty})(s))v(x_{\infty})\|,
\end{aligned}$$ where $C(f,K)
\doteqdot\sup_{(x,s)\in X\times K}$. Hence $(d)$ implies $(b)$.

\[21031238\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure, $Y_{0}\subset Y$ and ${\mathcal{V}}\in\prod_{x\in X}{\mathfrak{M}}_{x}$. We say that ${\mathcal{V}}$ is equicontinuous on $Y_{0}$ iff $(\forall j\in J)
(\exists a>0)
(\exists\,j_{1}\in J)
(\forall z\in X)
(\forall v_{z}\in{\mathfrak{E}}_{z})
$ $$\label{15422103}
\sup_{t\in Y_{0}}
\nu_{j}\left({\mathcal{V}}(z)(t)v_{z}\right)
\leq
a
\nu_{j_{1}}(v_{z})$$

While ${\mathcal{V}}$ is *equicontinuous* iff it is equicontinuous on $Y$. Finally ${\mathcal{V}}$ is *pointwise equicontinuous* iff it is equicontinuous on every point of $Y$ and *compactly equicontinuous* iff it is equicontinuous on every compact of $Y$. Note that in case ${\mathfrak{V}}$ is trivial with costant stalk $E$ then ${\mathcal{V}}$ is equicontinuous on $Y_{0}$ if and only if it is equicontinuous in the standard sense the following set of maps $\{
{\mathcal{V}}_{0}(z)(t)
\in{\mathcal{L}}(E)
\mid
(z,t)\in X\times Y_{0}
\}$, where ${\mathcal{V}}_{0}
\in
\left({\mathcal{L}}(E)^{Y}
\right)^{X}$ such that ${\mathcal{V}}(z)=(z,{\mathcal{V}}_{0}(z))$ for all $z\in X$.

\[20492003\] Let ${\mathfrak{V}}$ be trivial with costant stalk $E$, $A^{0}\in Bounded(E)$, $x_{\infty}\in X$ and [^13] $$\label{16062103}
\begin{cases}
{\mathcal{E}}_{0}
\subseteq
{\mathcal{C}_{b}
\left(X,E\right)}
\\
{\mathcal{E}}_{0}
\text{ equicontinuous 
set
at $x_{\infty}$}
\\
\{(X\ni x\mapsto a\in E)\mid a\in A^{0}\}
\subset
{\mathcal{E}}_{0}.
\end{cases}$$ Moreover let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that for all $x\in X$ $${\mathfrak{M}}_{x}
=
{\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{S_{x}}(\{x\}\times E)\right)}.$$ and $$\begin{cases}
{\mathcal{E}}
=\prod_{x\in X}
\{x\}\times{\mathcal{E}}_{0}
\\
\Theta
=
\{
B_{A^{0}}
\}
\end{cases}$$ where $B_{A^{0}}(x)
\doteqdot
\{x\}\times A^{0}$, then $$\label{23412003}
\begin{cases}
S_{x}
=
\{x\}\times A^{0},\forall x\in X
\\
{\mathfrak{M}}_{x}
\simeq
\{x\}\times 
{\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}.
\\
{\mathfrak{M}}
=
\bigcup_{x\in X}
{\mathfrak{M}}_{x}
\simeq
\bigcup_{x\in X}
\{x\}
\times
{\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}
\\
\prod_{x\in X}
{\mathfrak{M}}_{x}
\simeq
\prod_{x\in X}
\{x\}
\times
{\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}
\simeq
{\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}^{X}.
\end{cases}$$ If ${\mathfrak{W}}$ is full and $$\{
X\ni x\mapsto
\tau_{f}(x)
=(x,f)\in{\mathfrak{M}}_{x}
\mid 
f
\in
{\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}
\}
\subset
\Gamma(\rho),$$ then for all ${\mathcal{V}}\in\prod_{x\in X}^{b}{\mathfrak{M}}_{x}$, $(1)\Rightarrow(2)$ and $(3)\Leftrightarrow(4)$, where

1.  ${\mathcal{V}}\in\Gamma^{x_{\infty}}(\rho)$

2.  ${\mathcal{V}}_{0}
    \in{\mathcal{C}_{}
    \left(X,{\mathcal{C}_{c}
    \left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}\right)}$,

3.  ${\mathcal{V}}$ is compactly equicontinuous and ${\mathcal{V}}\in\Gamma^{x_{\infty}}(\rho)$

4.  ${\mathcal{V}}$ is compactly equicontinuous and ${\mathcal{V}}_{0}
    \in{\mathcal{C}_{}
    \left(X,{\mathcal{C}_{c}
    \left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}\right)}$.

Here in $(2)-(4)$ we consider the isomorphism $\prod_{x\in X}
{\mathfrak{M}}_{x}
\simeq
{\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}^{X}$, and set ${\mathcal{V}}_{0}
\in
{\mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}^{X}$ such that ${\mathcal{V}}(x)=(x,{\mathcal{V}}_{0}(x))$ for all $x\in X$.

For all $x\in X$ by ${\mathcal{B}}_{B_{A^{0}}}^{x}
=
\{(x,v_{0}(x))
\mid
v_{0}\in{\mathcal{E}}_{0},
v_{0}(X)\subseteq A^{0}\}
$ so ${\mathcal{B}}_{B_{A^{0}}}^{x}\subseteq A^{0}$. Moreover by construction $(X\ni x\mapsto a\in E)
\in{\mathcal{E}}_{0}$ for all $a\in A^{0}$, thus ${\mathcal{B}}_{B_{A^{0}}}^{x}= A^{0}$. Thus the first equality in follows, the others are trivial. By Proposition \[16572003\] $$\label{15102103}
(1)
\Leftrightarrow
\lim_{z\to x_{\infty}}
\sup_{t\in K}
\sup_{v_{0}\in
{\mathcal{E}}_{0}
\cap B_{A^{0}}
}
\nu_{j}\left((
{\mathcal{V}}_{0}(z)(t)-
{\mathcal{V}}_{0}(x_{\infty})(t))v_{0}(z)\right)
=0.$$ Moreover by construction we deduce that $\{(X\ni x\mapsto a\in E)\mid a\in A^{0}\}
\subset
{\mathcal{E}}_{0}
\cap B_{A^{0}}$, so $(2)$ follows by $(1)$ and . Let $v_{0}\in{\mathcal{E}}_{0}$ then for all $z\in X$ and $t\in Y$ $$\begin{aligned}
{1}
\label{15352103}
({\mathcal{V}}(z)(t)-
{\mathcal{V}}(x_{\infty})(t))
v_{0}(z)
&=
{\mathcal{V}}(z)(t)(v_{0}(z)-v_{0}(x_{\infty}))
+
\notag\\
({\mathcal{V}}(z)(t)-
{\mathcal{V}}(x_{\infty})(t))
v_{0}(x_{\infty})
&
+
{\mathcal{V}}(x_{\infty})(t)
(v_{0}(z)-v_{0}(x_{\infty})).\end{aligned}$$ Moreover by the hypothesis of equicontinuity at $x_{\infty}$ of the set ${\mathcal{E}}_{0}$, for all $j\in J$ $$\label{15542103}
\lim_{z\to x_{\infty}}
\sup_{v_{0}\in{\mathcal{E}}_{0}}
\nu_{j}(v_{0}(z)-v_{0}(x_{\infty}))
=0.$$ By and for all $j\in J$ there exists $j_{1}\in J$ and $a>0$ such that for all $z\in X$ $$\begin{aligned}
{1}
\label{15562103}
&
\sup_{t\in K}
\sup_{v_{0}\in
{\mathcal{E}}_{0}
\cap B_{A^{0}}
}
\nu_{j}\left((
{\mathcal{V}}_{0}(z)(t)-
{\mathcal{V}}_{0}(x_{\infty})(t))v_{0}(z)\right)
\leq
\notag\\
&
2a
\sup_{v_{0}\in
{\mathcal{E}}_{0}
\cap B_{A^{0}}
}
\nu_{j_{1}}
\left(v_{0}(z)-v_{0}(x_{\infty})\right)
+
\notag\\
&
\sup_{t\in K}
\sup_{v_{0}\in
{\mathcal{E}}_{0}
\cap B_{A^{0}}
}
\nu_{j}\left(
{\mathcal{V}}(z)(t)-
{\mathcal{V}}(x_{\infty})(t)\right)
v_{0}(x_{\infty}).\end{aligned}$$ Therefore by , and by $(4)$ follows $$\lim_{z\to x_{\infty}}
\sup_{t\in K}
\sup_{v_{0}\in
{\mathcal{E}}_{0}
\cap B_{A^{0}}
}
\nu_{j}\left((
{\mathcal{V}}_{0}(z)(t)-
{\mathcal{V}}_{0}(x_{\infty})(t))v_{0}(z)\right)
=0.$$ Hence $(1)$ follows by .

Main Claim
----------

The following is the preparatory definition for the first main structure of the paper.

\[ $
Graph(\left({\mathfrak{V}}_{1},{\mathfrak{V}}_{2}\right)
$ \] \[12432110bis\] Let $
{\mathfrak{V}}_{i}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}}_{i},\tau_{i}{\right\rangle}},\pi_{i},X,{\mathfrak{N}}_{i}{\right\rangle}}_{i=1}^{2}
$ be a couple of bunldes of $\Omega-$spaces and with the notations used in Remark \[21262110\] let $
\bigoplus_{i=1}^{2}
{\mathfrak{V}}_{i}
=
{\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}^{\oplus}),\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}){\right\rangle}},\pi_{{\mathbf{E}}^{\oplus}},X,{\mathfrak{n}}^{\oplus}{\right\rangle}}
$ be the bundle direct sum of the family $
\{
{\mathfrak{V}}_{i}
\}_{i=1}^{2}
$. Then we say that ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}$ is a *graph section* or $
{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}
\in
Graph\left(\left(
{\mathfrak{V}}_{1},{\mathfrak{V}}_{2}
\right)\right)
$ if

1.  $
    {\mathcal{T}}
    \in
    \prod_{x\in X}
    Graph(\left({\mathfrak{E}}_{1}\right)_{x}
    \times
    \left({\mathfrak{E}}_{2}\right)_{x})
    $;

2.  $x_{\infty}\in X$;

3.  $$\boxed{
    \Phi
    \subseteq
    \Gamma^{x_{\infty}}(\pi_{{\mathbf{E}}^{\oplus}})
    }$$ $\Phi$ is a linear space such that

    Graph inclusion

    :   $
        (\forall x\in X)
        (\forall \phi\in\Phi)
        (\phi(x)\in{\mathcal{T}}(x))
        $

    Asymptotic Graph

    :   $$\label{19350512bis}
        \boxed{
        \left\{
        \phi(x_{\infty})
        \mid
        \phi
        \in
        \Phi
        \right\}
        =
        {\mathcal{T}}(x_{\infty})
        }.$$

\[ $
Pregraph(\left({\mathfrak{V}}_{1},{\mathfrak{V}}_{2}\right)
$ \] \[16161212bis\] Let $
{\mathfrak{V}}_{i}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}}_{i},\tau_{i}{\right\rangle}},\pi_{i},X,{\mathfrak{N}}_{i}{\right\rangle}}_{i=1}^{2}
$ be a couple of bunldes of $\Omega-$spaces and $
\bigoplus_{i=1}^{2}
{\mathfrak{V}}_{i}
=
{\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}^{\oplus}),\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}){\right\rangle}},\pi_{{\mathbf{E}}^{\oplus}},X,{\mathfrak{n}}^{\oplus}{\right\rangle}}
$ the bundle direct sum of the family $
\{
{\mathfrak{V}}_{i}
\}_{i=1}^{2}
$. Then we say that ${\left\langle {\mathcal{T}}_{0},x_{\infty},\Phi{\right\rangle}}$ is a *pregraph section* or $
{\left\langle {\mathcal{T}}_{0},x_{\infty},\Phi{\right\rangle}}
\in
Pregraph\left(
{\mathfrak{V}}_{1},{\mathfrak{V}}_{2}
\right)
$ if

1.  $x_{\infty}\in X$;

2.  $
    {\mathcal{T}}_{0}
    \in
    \prod_{x\in X-\{x_{\infty}\}}
    Graph(\left({\mathfrak{E}}_{1}\right)_{x}
    \times
    \left({\mathfrak{E}}_{2}\right)_{x})
    $;

3.  $
    \Phi
    \subseteq
    \Gamma^{x_{\infty}}(\pi_{{\mathbf{E}}^{\oplus}})
    $ such that $\Phi$ is a linear space and $(\forall x\in X-\{x_{\infty}\})
    (\forall \phi\in\Phi)
    (\phi(x)\in{\mathcal{T}}_{0}(x))$.

We shall see in Lemma \[13001512b\] that it is possible to construct by any pregraph section $
{\left\langle {\mathcal{T}}_{0},x_{\infty},\Phi{\right\rangle}}
$ with suitable properties, a corresponding graph section ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}$ such that ${\mathcal{T}}{\upharpoonright}(X-\{x_{0}\})={\mathcal{T}}_{0}$, while ${\mathcal{T}}(x_{\infty})$ is defined by . To this end it is sufficient to show that ${\mathcal{T}}(x_{\infty})
\in
Graph({\mathfrak{E}}_{x_{\infty}}
\times{\mathfrak{E}}_{x_{\infty}})$.

\[23112110bis\] Notice that the fundamental requirement that any $\phi\in\Phi$ is a selection continuous in $x_{\infty}$ implies that $$\begin{cases}
\left\{
\lim_{z\to x_{\infty}}
\phi(z)
\mid
\phi\in\Phi
\right\}
=
{\mathcal{T}}(x_{\infty})
\in 
Graph(\left({\mathfrak{E}}_{1}\right)_{x_{\infty}}
\times
\left({\mathfrak{E}}_{2}\right)_{x_{\infty}})
\\
\text{with}
\\
\phi(z)\in
{\mathcal{T}}(z)
\in 
Graph(\left({\mathfrak{E}}_{1}\right)_{z}
\times
\left({\mathfrak{E}}_{2}\right)_{z}),\,
\forall z\in X-\{x_{\infty}\},
\end{cases}$$ which justify the name of asymptotic graph given to . Moreover by setting $X\ni z\mapsto
\phi_{i}(z)
\doteqdot
\Pr_{i}^{z}(\phi(z))$ we have by Corollary \[17571212\] we have for all $i=1,2$ $$\begin{cases}
\label{09382412bis}
\left\{
\lim_{z\to x_{\infty}}
\phi_{i}(z)
\mid
\phi\in\Phi
\right\}
=
\Pr_{i}^{x_{\infty}}({\mathcal{T}}(x_{\infty}))
\text{with}
\\
\phi(z)\in
{\mathcal{T}}(z)
\in 
Graph(\left({\mathfrak{E}}_{1}\right)_{z}
\times
\left({\mathfrak{E}}_{2}\right)_{z}),\,
\forall z\in X-\{x_{\infty}\}.
\end{cases}$$ Finally for $i=1,2$ by Corollary \[17571212\] and Corollary \[28111707\] we have $(1_{i})
\Leftrightarrow
(2_{i})$

$(1_{i})$

:   $(\exists\,
    \sigma\in\Gamma(\pi))
    (\sigma(x_{\infty})
    =\phi_{i}(x_{\infty}))$ such that $$(\forall j\in J)
    (\lim_{z\to x_{\infty}}
    \nu_{j}(\phi_{i}(z)-\sigma(z))=0);$$

$(1_{ii})$

:   $(\forall\sigma\in\Gamma(\pi)
    \mid
    \sigma(x_{\infty})
    =\phi_{i}(x_{\infty}))$ we have $$(\forall j\in J)
    (\lim_{
    z\to x_{\infty}}
    \nu_{j}(\phi_{i}(z)-\sigma(z))=0).$$

\[ Class $
\Delta{\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}
$ \] \[15312011bis\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure and let us denote ${\mathfrak{D}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{B}},\gamma{\right\rangle}},\eta,X,{\mathfrak{L}}{\right\rangle}}$. Thus $\Omega\,\in
\Delta
{\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$ if

1.  $
    \Omega
    \subseteq
    Graph
    \left(
    \left(
    {\mathfrak{V}},{\mathfrak{V}}
    \right)
    \right)
    $;

2.  **Projector Section associated to ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}$:** $
    \forall
    {\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}
    \in
    \Omega
    $ $$\label{18270303}
    \boxed{
    \left(\exists\,
    {\mathcal{P}}
    \in
    \Gamma^{x_{\infty}}(\eta)
    \cap
    \prod_{x\in X}
    \Pr({\mathfrak{E}}_{x})
    \right)
    \left(\forall x\in X\right)
    \left({\mathcal{P}}(x)
    T_{x}
    \subseteq
    T_{x}
    {\mathcal{P}}(x)\right).
    }$$

Here $
T_{x}:
D_{x}
\subseteq
{\mathfrak{E}}_{x}
\to
{\mathfrak{E}}_{x}
$ is the map such that ${\mathcal{T}}(x)=Graph(T_{x})$, for all $x\in X$.

\[MAIN\] \[19520412bis\] Under the assumptions in Definition \[15312011bis\], possibly with ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ invariant, find elements in the class $$\Delta{\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}.$$

Induction of Sections of Semigroups 
------------------------------------

\[ Induction of Sections of Semigroups \] \[19490412bis\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that $\{{\mathfrak{E}}_{x}\}_{x\in X}$ is a family of sequentially complete Hlcs and ${\mathbf{U}}({\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x}))
\subset
{\mathfrak{M}}_{x}$, where we set $
{\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}
$ and ${\mathfrak{W}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$. Then $\Omega\in
\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$ if

1.  $
    \Omega
    \subseteq
    Graph(\left({\mathfrak{E}},{\mathfrak{E}})\right)
    $;

2.  **Semigroup Section associated to ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}$:**\
    $
    \forall
    {\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}
    \in
    \Omega
    $ $$\boxed{
    \exists\,
    {\mathcal{U}}_{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}}
    \in
    \Gamma^{x_{\infty}}(\rho)
    }$$ such that $\forall x\in X$

    1.  $
        {\mathcal{U}}_{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}}(x)
        $ is an equicontinuous $(C_{0})-$semigroup on ${\mathfrak{E}}_{x}$;

    2.  $
        (\forall x\in X)
        ({\mathcal{T}}(x)=Graph(R_{x}))
        $.

Here $R_{x}$ is the infinitesimal generator of the semigroup $
{\mathcal{U}}_{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}}(x)
\in
{\mathcal{C}_{c}
\left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}
({\mathfrak{E}}_{x})\right)}
$.

\[20090412bis\] Under the assumptions in Definition \[19490412bis\], possibly with ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ compatible, find elements in the class $$\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}.$$

\[20430412bis\] Notice that $
\forall
{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}
\in
\Omega
$ there exists only one semigroup section associated to it. Moreover ${\mathcal{U}}_{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}}$ is characterized by any of the equivalent conditions in Lemma \[15482712\] with $U=X$ and $Y={\mathbb {R}}^{+}$.

 Induction of a Section of Semigroups - Projectors  {#20110412bis}
---------------------------------------------------

\[ Induction of a Section of Semigroups - Projectors \] \[18550612bis\] Let $
{\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}
$, ${\mathfrak{D}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{B}},\gamma{\right\rangle}},\eta,X,{\mathfrak{L}}{\right\rangle}}$ and ${\mathfrak{W}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$. Moreover let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that $\{{\mathfrak{E}}_{x}\}_{x\in X}$ is a family of sequentially complete Hlcs and ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure. Then $\Psi
\in
\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}
$ if

1.  $\Psi\subseteq
    \bigcup_{x_{\infty}\in X}
    \Gamma^{x_{\infty}}(\rho)
    $;

2.  $
    (\forall{\mathcal{U}}\in\Psi)
    (\forall x\in X)
    $ $
    ({\mathcal{U}}(x)
    $ is an equicontinuous $(C_{0})-$semigroup on ${\mathfrak{E}}_{x})$;

3.  **Projector Section associated to ${\mathcal{U}}$:**\
    $(\forall x_{\infty}\in X)
    (\forall{\mathcal{U}}
    \in\Psi\cap\Gamma^{x_{\infty}}(\rho))$ $$\label{16221411bis}
    \boxed{
    \left(\exists\,
    {\mathcal{P}}
    \in
    \Gamma^{x_{\infty}}(\eta)
    \cap
    \prod_{x\in X}
    \Pr({\mathfrak{E}}_{x})
    \right)
    \left(\forall x\in X\right)
    \left(
    {\mathcal{P}}(x)H_{x}
    \subseteq
    H_{x}
    {\mathcal{P}}(x)
    \right).
    }$$

Here $H_{x}$ is the infinitesimal generator of the semigroup $
{\mathcal{U}}(x)
\in
{\mathcal{C}_{c}
\left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}
({\mathfrak{E}}_{x})\right)}$ for all $x\in X$.

\[S-P\] \[20290412bis\] Under the assumptions in Definition \[19490412bis\], possibly with ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ compatible and ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ invariant, find elements in the class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}.$

Claims \[20090412bis\] and \[20290412bis\] can be used to solve the Main claim \[19520412bis\] indeed

\[20340412bis\] Let $
{\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}
$, ${\mathfrak{D}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{B}},\gamma{\right\rangle}},\eta,X,{\mathfrak{L}}{\right\rangle}}$ and ${\mathfrak{W}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$. Moreover let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that $\{{\mathfrak{E}}_{x}\}_{x\in X}$ is a family of sequentially complete Hlcs and ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure. Assume that

1.  $
    \Omega\in
    \Omega\in
    \Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}
    $;

2.  $
    \Psi
    \in
    \Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}
    $;

and $$\label{20500412bis}
\boxed{
(\forall
{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}
\in\Omega)
(
{\mathcal{U}}_{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}}
\in
\Psi).
}$$ Then $$\Omega\,\in\Delta
{\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}},$$ i.e. $\Omega$ satisfies the Main claim \[19520412bis\]. Moreover $$\left(
\forall
{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}
\in
\Omega
\right)
\left(
\exists\,
{\mathcal{P}}
\in
\Gamma^{x_{\infty}}(\eta)
\right)
\left(
\exists\,
{\mathcal{U}}
\in
\Gamma^{x_{\infty}}(\rho)
\right)$$

1.  $
    {\mathcal{U}}(x)
    $ is an equicontinuous $(C_{0})-$semigroup on ${\mathfrak{E}}_{x}$, for all $x\in X$;

2.  $
    \left(\forall x\in X\right)
    \left({\mathcal{P}}(x)
    \in
    \Pr({\mathfrak{E}}_{x})
    \right)
    $;

3.  $
    (\forall x\in X)
    ({\mathcal{T}}(x)=Graph(R_{x}))
    $;

4.  $
    \forall x\in X
    $ $${\mathcal{P}}(x)R_{x}
    \subseteq
    R_{x}
    {\mathcal{P}}(x).$$

Here $R_{x}$ is the infinitesimal generator of the semigroup $
{\mathcal{U}}(x)
\in
{\mathcal{C}_{c}
\left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)}$, for all $x\in X$.

Semigroup Approximation Theorems
================================

General Approximation Theorem I {#11542812}
-------------------------------

\[15411512b\] For any $
{\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}
$ bunlde of $\Omega-$spaces and any ${\left\langle {\mathcal{T}}_{0},x_{\infty},\Phi{\right\rangle}}
\in
Pregraph
\left(\left({\mathfrak{V}},{\mathfrak{V}}\right)\right)$, set $X_{0}\doteqdot X-\{x_{\infty}\}$, and for any $\phi\in\Phi$ $\phi_{i}(x)\doteqdot
\Pr_{i}^{x}(\phi(x))$ for all $x\in X$ and $i=1,2$. Moreover let us denote by $T_{x}$ the operator in ${\mathfrak{E}}_{x}$ such that $Graph(T_{x})={\mathcal{T}}_{0}(x)$, for all $x\in X_{0}$, while $
{\mathcal{T}}
\in
\prod_{x\in X}
Graph({\mathfrak{E}}_{x}
\times{\mathfrak{E}}_{x})
$ so that $$\begin{cases}
{\mathcal{T}}{\upharpoonright}X-\{x_{\infty}\}
\doteqdot
{\mathcal{T}}_{0}
\\
{\mathcal{T}}(x_{\infty})
\doteqdot
\{\phi(x_{\infty})\mid\phi\in\Phi\},
\end{cases}$$ in addition set $$D(T_{x_{\infty}})
\doteqdot
\Pr_{1}^{x_{\infty}}({\mathcal{T}}(x_{\infty}))
=
\{\phi_{1}(x_{\infty})
\mid\phi\in\Phi\}.$$ Finally for any map $F:A\to B$ set ${\mathcal{R}}(F)\doteqdot F(A)$ the range of $F$.

\[15401512b\] Let $
{\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}
$ be a bunlde of $\Omega-$spaces and ${\left\langle {\mathcal{T}}_{0},x_{\infty},\Phi{\right\rangle}}
\in
Pregraph
\left(\left({\mathfrak{V}},{\mathfrak{V}}\right)\right)$. By Corollary \[17571212\] $\forall\phi\in\Phi$ $$\label{15211512b}
\begin{cases}
\phi_{i}\in\Gamma^{x_{\infty}}(\pi), i=1,2\\
(\forall x\in X_{0})
(\phi_{2}(x)=T_{x}\phi_{1}(x)).
\end{cases}$$

\[13001512b\] Let $
{\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}
$ be a bundle of $\Omega-$spaces, where ${\mathfrak{N}}\doteqdot\{\nu_{j}\mid j\in J\}$. Moreover ${\left\langle {\mathcal{T}}_{0},x_{\infty},\Phi{\right\rangle}}
\in
Pregraph
\left(\left({\mathfrak{V}},{\mathfrak{V}}\right)\right)$. If for all $x\in X_{0}$, $v_{x}\in Dom(T_{x})$, $\lambda>0$ and $j\in J$ we have $\nu_{j}((\lambda- T_{x})v_{x})
\geq
\lambda\nu_{j}(v_{x})$ and $D(T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$, then $${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}
\in
Graph
\left(\left({\mathfrak{V}},{\mathfrak{V}}\right)\right).$$ Moreover the following $$\label{172212312b}
T_{x_{\infty}}:
D(T_{x_{\infty}})
\ni
\phi_{1}(x_{\infty})
\mapsto
\phi_{2}(x_{\infty})$$ is a well-defined linear operator in ${\mathfrak{E}}_{x_{\infty}}$ such that $
Graph(T_{x_{\infty}})
=
{\mathcal{T}}(x_{\infty})
$ and $\forall v_{x_{\infty}}\in 
Dom(T_{x_{\infty}})$, $\forall\lambda>0$ and $\forall j\in J$ we have $$\nu_{j}((\lambda- T_{x_{\infty}})
v_{x_{\infty}})
\geq
\lambda\nu_{j}(v_{x_{\infty}}).$$

Clearly $
{\mathcal{T}}(x_{\infty})
\in
Graph({\mathfrak{E}}_{x_{\infty}}
\times{\mathfrak{E}}_{x_{\infty}})
$ if and only if $\phi_{1}(x_{\infty})
={\mathbf{0}}_{x_{\infty}}
$ implies $\phi_{2}(x_{\infty})={\mathbf{0}}_{x_{\infty}}$, $
\forall\phi\in\Phi
$, moreover denoting by $T_{x_{\infty}}$ the corresponding operator we have that $T_{x_{\infty}}:
D(T_{x_{\infty}})\to{\mathfrak{E}}_{x_{\infty}}$ is a linear operator. Any real map $F$ defined on a topological space is l.s.c. at a point iff $-F$ is u.s.c. at the same point, see [@BourGT $\S 6.2.$ Ch.$4$], thus by [@BourGT Prop. $3$ $\S 6.2.$ Ch.$4$] and [@BourGT (13),$\S 5.6.$ Ch.$4$] $F:X\to{\mathbb {R}}$ is u.s.c. in $a\in X$ iff $\varlimsup_{x\to a}F(x)=F(a)$. Moreover by [@BourGT $\S 6.2.$ Ch.$4$] we know that $F:X\to{\overline}{{\mathbb {R}}}$ is l.s.c. at $a$ iff $F$ is continuous at $a$ providing ${\overline}{{\mathbb {R}}}$ with the following topology $\{\emptyset,[-\infty,\infty],]a,\infty[
\mid a\in{\mathbb {R}}\}$. Thus for any map $\sigma:Y\to X$ continuous at $b$ such that $\sigma(b)=a$ we have that $F\circ\sigma$ is l.s.c. at $a$. Hence because $(-F)\circ\sigma=-(F\circ\sigma)$ we can state that if $F:X\to{\overline}{{\mathbb {R}}}$ is u.s.c. at $a$ then for any map $\sigma:Y\to X$ continuous at $b$ such that $\sigma(b)=a$ we have that $F\circ\sigma$ is u.s.c. at $a$. Therefore by using [@gie $1.6.(ii)$] we have $\forall\sigma\in\Gamma^{x_{\infty}}(\pi)$ and $\forall j\in J$ $$\label{16051512b}
\nu_{j}(\sigma(x_{\infty}))
=
\varlimsup_{x\to x_{\infty}}
\nu_{j}(\sigma(x)).$$ Let $\psi\in\Phi$ such that $\psi_{1}(x_{\infty})={\mathbf{0}}_{x_{\infty}}$ thus $\forall\phi\in\Phi$, $\forall\lambda>0$, $\forall x\in X_{0}$ and $\forall j\in J$ we have by and $$\begin{aligned}
{2}
\label{16401512b}
\nu_{j}\left(
\lambda\phi_{1}(x_{\infty})-
\phi_{2}(x_{\infty})-
\lambda\psi_{2}(x_{\infty})
\right)
&
=
\notag
\\
\varlimsup_{x\to x_{\infty}}
\nu_{j}\left((\lambda-T_{x})
(\phi_{1}(x)+\lambda\psi_{1}(x))\right)
&
\geq
\notag
\\
\varlimsup_{x\to x_{\infty}}
\lambda
\nu_{j}\left(\phi_{1}(x)+
\lambda\psi_{1}(x))\right)
&
=
\lambda
\nu_{j}(\phi_{1}(x_{\infty})),\end{aligned}$$ where, the inequality comes by [@BourGT Prop. $11$ $\S 5.6.$ Ch.$4$]) and by the hypothesis $
\nu_{j}\left((\lambda-T_{x})
(\phi_{1}(x)+\lambda\psi_{1}(x))\right)
\geq
\lambda
\nu_{j}\left((\phi_{1}(x)+
\lambda\psi_{1}(x))\right)
$ for all $x\in X_{0}$. Now $
\lim_{\lambda\to\infty}v/\lambda
=
{\mathbf{0}}_{x_{\infty}}
$ for any $v\in{\mathfrak{E}}_{x_{\infty}}$, hence by the fact that $\nu_{j}^{x_{\infty}}
\doteqdot
\nu_{j}{\upharpoonright}{\mathfrak{E}}_{x_{\infty}}$ is a continuous seminorm and by $
(\forall j\in J)
(\forall\phi\in\Phi)
$ $$\label{17011512b}
\nu_{j}\left(
\phi_{1}(x_{\infty})-\psi_{2}(x_{\infty})
\right)
=
\lim_{\lambda\to\infty}
\frac{\nu_{j}\left(
\lambda\phi_{1}(x_{\infty})-
\phi_{2}(x_{\infty})-
\lambda\psi_{2}(x_{\infty})
\right)}{\lambda}
\geq
\nu_{j}(\phi_{1}(x_{\infty})).$$ By hypothesis $
D(T_{x_{\infty}})
=
\{\phi_{1}(x_{\infty})
\mid\phi\in{\mathcal{T}}(x_{\infty})\}
$ is dense in ${\mathfrak{E}}_{x_{\infty}}$ thus $\nu_{j}(\psi_{2}(x_{\infty}))=0$ for all $j\in J$. Indeed let $j\in J$ and $v\in{\mathfrak{E}}_{x_{\infty}}$ thus $\exists\,\{\phi^{\alpha}\}_{\alpha\in D}$ net in $\Phi$ such that $
\lim_{\alpha\in D}
\phi_{1}^{\alpha}(x_{\infty})
=
v
$ in ${\mathfrak{E}}_{x_{\infty}}$. So by the continuity of $\nu_{j}^{x_{\infty}}$ and by we have $\forall v\in{\mathfrak{E}}_{x_{\infty}}$ $$\nu_{j}\left(
v
-\psi_{2}(x_{\infty})
\right)
=
\lim_{\alpha\in D}
\nu_{j}\left(
\phi_{1}^{\alpha}(x_{\infty})
-\psi_{2}(x_{\infty})
\right)
\geq
\lim_{\alpha\in D}
\nu_{j}\left(
\phi_{1}^{\alpha}(x_{\infty}))
\right)
=
\nu_{j}(v).$$ True in particular for $v=3\psi_{2}(x_{\infty})$, which implies $\nu_{j}\left(\psi_{2}(x_{\infty})\right)
=0$. Hence $\psi_{2}(x_{\infty})={\mathbf{0}}_{x_{\infty}}$ because of ${\mathfrak{E}}_{x_{\infty}}$ is a Hausdorff lcs for which $\{\nu_{j}^{x_{\infty}}\}_{j\in J}$ is a generating set of seminorms of its topology. Thus $T_{x_{\infty}}$ is a well-defined (necessarly linear) operator in ${\mathfrak{E}}_{x_{\infty}}$ and consequently ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}
\in
Graph
\left(\left({\mathfrak{V}},{\mathfrak{V}}\right)\right)$. Finally $(\forall j\in J)
(\forall\phi\in\Phi)
(\forall\lambda>0)$ $$\begin{aligned}
{1}
\nu_{j}
((\lambda-T_{x_{\infty}})\phi_{1}(x_{\infty}))
&=
\\
\nu_{j}
(\lambda\phi_{1}(x_{\infty})-
\phi_{2}(x_{\infty}))
&=
\qquad
\text{ by }
\eqref{15211512b},
\eqref{16051512b}
\\
\varlimsup_{x\to x_{\infty}}
\nu_{j}
(\lambda\phi_{1}(x)-
\phi_{2}(x))
&=
\qquad
\text{ by }
\eqref{15211512b}
\\
\varlimsup_{x\to x_{\infty}}
\nu_{j}
((\lambda-T_{x})\phi_{1}(x))
&\geq
\qquad
\text{ by hypoth. 
and 
\cite[Prop. $11$ $\S 5.6.$ Ch.$4$]{BourGT})
}
\\
\varlimsup_{x\to x_{\infty}}
\nu_{j}
(\lambda\phi_{1}(x))
&=
\nu_{j}
(\lambda\phi_{1}(x_{\infty})).\end{aligned}$$

\[161172901\] In addition to the hypotheses and notations of Lemma \[13001512b\] assume that $(\forall x\in X_{0})
(\forall\lambda\in{\mathbb {R}})
(\forall j\in J)
(\forall v_{x}\in Dom(T_{x}))
$ $$\label{16292901}
\nu_{j}
(({\mathbf{1}}-\lambda T_{x})v_{x})
\geq
\nu_{j}(v_{x}).$$ Thus $(\forall\lambda\in{\mathbb {R}})
(\forall j\in J)
(\forall v_{x_{\infty}}\in 
Dom(T_{x_{\infty}}))$ $$\label{20022901}
\nu_{j}(({\mathbf{1}}-\lambda T_{x_{\infty}})
v_{x_{\infty}})
\geq
\nu_{j}(v_{x_{\infty}}).$$ Moreover $\forall\lambda\in{\mathbb {R}}$ $$\label{20102901}
\begin{cases}
\exists\,
({\mathbf{1}}-\lambda T_{x_{\infty}})^{-1}
\in{\mathcal{L}}
({\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}}),
{\mathfrak{E}}_{x_{\infty}}),
\\
(\forall w\in
{\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}}))
(\forall j\in J)
\nu_{j}(({\mathbf{1}}-\lambda T_{x_{\infty}})^{-1}w)
\leq\nu_{j}(w).
\end{cases}$$ Finally $$\label{20342901}
{\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}})
\text{ is closed in ${\mathfrak{E}}_{x_{\infty}}$.}$$

$(\forall j\in J)
(\forall\phi\in\Phi)
(\forall\lambda\in{\mathbb {R}})$ $$\begin{aligned}
{1}
\nu_{j}
(({\mathbf{1}}-
\lambda T_{x_{\infty}})\phi_{1}(x_{\infty}))
&=
\\
\nu_{j}
(\phi_{1}(x_{\infty})-
\lambda
\phi_{2}(x_{\infty}))
&=
\qquad
\text{ by }
\eqref{15211512b},
\eqref{16051512b}
\\
\varlimsup_{x\to x_{\infty}}
\nu_{j}
(\phi_{1}(x)-
\lambda\phi_{2}(x))
&=
\qquad
\text{ by }
\eqref{15211512b}
\\
\varlimsup_{x\to x_{\infty}}
\nu_{j}
(({\mathbf{1}}-\lambda T_{x})\phi_{1}(x))
&\geq
\qquad
\text{ by \eqref{16292901} 
and 
\cite[Prop. $11$ $\S 5.6.$ Ch.$4$]{BourGT})
}
\\
\varlimsup_{x\to x_{\infty}}
\nu_{j}
(\phi_{1}(x))
&=
\nu_{j}
(\phi_{1}(x_{\infty})).\end{aligned}$$ thus follows. Let $\lambda\in{\mathbb {R}}$, by we obtain , indeed $\forall f,g\in Dom(T_{x_{\infty}})$ such that $({\mathbf{1}}-\lambda T_{x_{\infty}})f
=
({\mathbf{1}}-\lambda T_{x_{\infty}})g$ we have $\forall j\in J$ $$0
=
\nu_{j}(({\mathbf{1}}-\lambda T_{x_{\infty}})
(f-g))
\geq
\nu_{j}(f-g),$$ so $f=g$ because of by construction ${\mathfrak{E}}_{x_{\infty}}$ is Hausdorff. Thus the following is a well-set map $$({\mathbf{1}}-\lambda T_{x_{\infty}})^{-1}:
{\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}})
\ni
({\mathbf{1}}-\lambda T_{x_{\infty}})f
\mapsto
f
\in
{\mathfrak{E}}_{x_{\infty}},$$ moreover by we obtain the second sentence of , hence the first one follows by the fact that the inverse map of any linear operator is linear. By , [@BourGT Prop. $3$ $\S 3.1.$ Ch.$3$] and [@BourGT Prop. $11$ $\S 3.6.$ Ch.$2$] we deduce that $$\label{20182901}
(\exists\,!\, 
B\in{\mathcal{L}}\left(
{\overline}{{\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}})},
{\mathfrak{E}}_{x_{\infty}}\right))
(B{\upharpoonright}{\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}})
=
({\mathbf{1}}-\lambda T_{x_{\infty}})^{-1}).$$ Let $w\in
{\overline}{{\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}})}$ thus $\exists\,\{f_{\alpha}\}_{\alpha\in D}$ net in $Dom(T_{x_{\infty}})$ such that $$\label{19252901}
w=\lim_{\alpha\in D}
({\mathbf{1}}-\lambda T_{x_{\infty}})
f_{\alpha},$$ therefore by $$\label{19512901a}
Bw=
\lim_{\alpha\in D}
f_{\alpha},$$ while by and $$\begin{aligned}
{2}
w-Bw
&=
\lim_{\alpha\in D}
\left(
(f_{\alpha}-\lambda T_{x_{\infty}}f_{\alpha})
-f_{\alpha}\right)
\\
&=
\lim_{\alpha\in D}
-\lambda T_{x_{\infty}}f_{\alpha}.\end{aligned}$$ So $$\label{19512901b}
Bw-w
=
\lim_{\alpha\in D}
\lambda T_{x_{\infty}}f_{\alpha}.$$ By , and the fact that $\lambda T_{x_{\infty}}$ is closed, we obtain $$\begin{cases}
Bw\in Dom(T_{x_{\infty}}),
\\
\lambda T_{x_{\infty}}(Bw)
=
Bw-w,
\end{cases}$$ which means $w=({\mathbf{1}}-\lambda T_{x_{\infty}})
Bw$, so $w\in {\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}})$ and follows.

\[14253001\] Let us assume the hypotheses of Lemma \[161172901\], moreover let $\lambda\in
{\mathbb {R}}-\{0\}$, $\{\lambda_{n}\}_{n\in{\mathbb{N}}}\subset{\mathbb {R}}-\{0\}$ such that $\lim_{{n\in{\mathbb{N}}}}\lambda_{n}=\lambda$. Thus $$\bigcap_{n\in{\mathbb{N}}}
{\mathcal{R}}({\mathbf{1}}-\lambda_{n}T_{x_{\infty}})
\subseteq
{\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}}).$$

Set only in this proof $T\doteqdot T_{x_{\infty}}$. Let $n\in{\mathbb{N}}$, by $\exists\,({\mathbf{1}}-\lambda_{n}T)^{-1}:
{\mathcal{R}}({\mathbf{1}}-\lambda_{n}T_{x_{\infty}})
\to
Dom(t)$ moreover $$\begin{cases}
{\mathbf{1}}-\lambda T
=
\lambda(\lambda^{-1}-T),
\\
({\mathbf{1}}-\lambda_{n}T)^{-1}
=
\lambda_{n}^{-1}
(\lambda_{n}^{-1}-T)^{-1}.
\end{cases}$$ Let $g\in\bigcap_{n\in{\mathbb{N}}}
{\mathcal{R}}({\mathbf{1}}-\lambda_{n}T_{x_{\infty}})$ thus $$\begin{aligned}
{2}
({\mathbf{1}}-\lambda T)
({\mathbf{1}}-\lambda_{n} T)^{-1}g
-
g
&=
\frac{\lambda}{\lambda_{n}}
(\lambda^{-1}-T)
(\lambda_{n}^{-1}-T)^{-1}g
-
g
\\
&=
\frac{\lambda}{\lambda_{n}}
\left(
\lambda^{-1}
(\lambda_{n}^{-1}-T)^{-1}g
-
\lambda_{n}^{-1}
(\lambda_{n}^{-1}-T)^{-1}g
\right)
\\
&=
\frac{\lambda}{\lambda_{n}}
(\lambda^{-1}-
\lambda_{n}^{-1})
(\lambda_{n}^{-1}-T)^{-1}g,\end{aligned}$$ where in the second equality we considered that $-T
(\lambda_{n}^{-1}-T)^{-1}g-g
=
-\lambda_{n}^{-1}
(\lambda_{n}^{-1}-T)^{-1}
g$ obtained by $(\lambda_{n}^{-1}-T)
(\lambda_{n}^{-1}-T)^{-1}g
=g$. Thus $\forall j\in J$ by $$\nu_{j}
\left(
({\mathbf{1}}-\lambda T)
({\mathbf{1}}-\lambda_{n} T)^{-1}g
-
g\right)
\leq
\left|
\frac{\lambda}{\lambda_{n}}
\right|
|\lambda^{-1}-\lambda_{n}^{-1}|
\nu_{j}(g).$$ But $\lim_{n\in{\mathbb{N}}}
|\lambda^{-1}-\lambda_{n}^{-1}|
=1$ and $\lim_{n\in{\mathbb{N}}}
|\lambda^{-1}-\lambda_{n}^{-1}|
=0$ so $\nu_{j}
\left(
({\mathbf{1}}-\lambda T)
({\mathbf{1}}-\lambda_{n} T)^{-1}g
-
g
\right)=0$, for all $j\in J$. Therefore $$\lim_{n\in{\mathbb{N}}}
({\mathbf{1}}-\lambda T)
({\mathbf{1}}-\lambda_{n} T)^{-1}g
=
g,$$ and the statement follows by .

\[17552901\] Under the hypotheses and notations of Lemma \[13001512b\] we have that ${\mathbf{1}}-\lambda T_{x_{\infty}}$ is a closed operator.

Let $(a,b)\in
{\overline}{Graph({\mathbf{1}}-\lambda T_{x_{\infty}})}$ closure in the space ${\mathfrak{E}}_{x_{\infty}}
\times
{\mathfrak{E}}_{x_{\infty}}$ with the product topology. Thus $(\forall{\varepsilon}>0)
(\forall j\in J)
(\exists\,v_{({\varepsilon},j)}\in Dom(T_{x_{\infty}}))$ $$\begin{cases}
\nu_{j}
(a-v_{({\varepsilon},j)})
<\frac{{\varepsilon}}{2},
\\
\nu_{j}
(b-({\mathbf{1}}-\lambda T_{x_{\infty}})v_{({\varepsilon},j)})
<\frac{{\varepsilon}}{2},
\end{cases}$$ so $$\nu_{j}
((b-a)+\lambda T_{x_{\infty}}v_{({\varepsilon},j)})
\leq
\nu_{j}(b-({\mathbf{1}}-\lambda T_{x_{\infty}})
v_{({\varepsilon},j)})
+
\nu_{j}(a-v_{({\varepsilon},j)})
\leq
{\varepsilon}.$$ Therefore $(\forall{\varepsilon}>0)
(\forall j\in J)
(\exists\,v_{({\varepsilon},j)}\in Dom(T_{x_{\infty}}))$ $$\begin{cases}
\nu_{j}
(a-v_{({\varepsilon},j)})
<{\varepsilon},
\\
\nu_{j}
\left(
(b-a)-(-\lambda T_{x_{\infty}}v_{({\varepsilon},j)})
\right),
\end{cases}$$ which means $(a,(b-a))
\in{\overline}{Graph(-\lambda T_{x_{\infty}})}$. Moreover $-\lambda T_{x_{\infty}}$ is a closed operator thus $b-a=-\lambda T_{x_{\infty}}a$ or equivalently $(a,b)\in 
Graph({\mathbf{1}}-\lambda T_{x_{\infty}})$.

\[18581512b\] By we have $\forall\phi\in\Phi$ that $
\phi_{1}(x_{\infty})
=
\lim_{z\to x_{\infty}}
\phi_{1}(z)
$ and $
\phi_{2}(x_{\infty})
=
\lim_{z\to x_{\infty}}
\phi_{2}(z)
=
\lim_{z\to x_{\infty}}
T_{x}\phi_{1}(z)
$, hence $$\begin{cases}
\phi_{1}(x_{\infty})
=
\lim_{z\to x_{\infty}}
\phi_{1}(z)
\\
T_{x_{\infty}}
\phi_{1}(x_{\infty})
=
\lim_{z\to x_{\infty}}
T_{z}
\phi_{1}(z).
\end{cases}$$

\[15062301\] Let $\lambda\in{\mathbb {R}}^{+}$ set $$\mu_{\lambda}:
{\mathcal{C}_{cs}
\left({\mathbb {R}}^{+},{\mathbb {R}}\right)}
\ni
f
\mapsto
\int_{{\mathbb {R}}^{+}}
e^{-s\lambda}
f(s)\,ds,$$ where the integral is with respect to the Lebesgue measure on ${\mathbb {R}}^{+}$.

\[16401812b\] Let ${\mathfrak{W}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ and ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that for all $x\in X$ $$\label{18470109}
{\mathfrak{M}}_{x}
\subseteq
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},
{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\mu_{\lambda}),$$ about $S_{x}$ and ${\mathfrak{E}}_{x}$ see Definition \[10282712\]. Let $x\in X$, ${\mathcal{O}}\subseteq\Gamma(\rho)$. and ${\mathcal{D}}\subseteq\Gamma(\pi)$. By recalling Def. \[15012602\], we say that ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the *Laplace duality property on ${\mathcal{O}}$ and ${\mathcal{D}}$ at $x$*, shortly ${\mathbf{LD}}_{x}({\mathcal{O}},{\mathcal{D}})$ if $$(\forall\lambda>0)
({\mathfrak{L}}(\Gamma_{{\mathcal{O}}}^{x}(\rho))_{\lambda}
\bullet
\Gamma_{{\mathcal{D}}}^{x}(\pi)
\subseteq
\Gamma^{x}(\pi)).$$ Moreover we say that ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the *full Laplace duality property on ${\mathcal{O}}$ and ${\mathcal{D}}$*, shortly ${\mathbf{LD}}({\mathcal{O}},{\mathcal{D}})$ if $$(\forall\lambda>0)
({\mathfrak{L}}({\mathcal{O}})_{\lambda}
\bullet
{\mathcal{D}}
\subseteq
\Gamma(\pi)).$$ Finally ${\mathbf{LD}}$ is for ${\mathbf{LD}}(\Gamma(\rho),\Gamma(\pi))$. Here ${\mathfrak{L}}:
\prod_{x\in X}
{\mathfrak{M}}_{x}
\to
\prod_{x\in X}
{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})^{{\mathbb {R}}^{+}}$ such that $(\forall x\in X)
(\forall\lambda\in{\mathbb {R}}^{+})$ $${\mathfrak{L}}(F)(x)(\lambda)
\doteqdot
\int_{0}^{\infty}
e^{-\lambda s}
F(x)(s)\,ds,$$ where we recall that the integration is with respect to the Lebesgue measure on ${\mathbb {R}}^{+}$ and with respect to the lct on ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$. Finally we used the notations in Def \[15012602\].

\[10412505\] Under the notations and assumptions of Def. \[16401812b\] we have $$\begin{cases}
{\mathfrak{L}}(\Gamma_{{\mathcal{O}}}^{x}(\rho))
\subseteq
\Gamma_{{\mathcal{O}}}^{x}(\rho)
\\
(\forall t>0)
(\Gamma_{{\mathcal{O}}}^{x}(\rho)_{t}
\bullet
\Gamma_{{\mathcal{D}}}^{x}(\pi)
\subseteq
\Gamma^{x}(\pi))
\end{cases}
\Rightarrow
{\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}
\text{has the }
{\mathbf{LD}}_{x}({\mathcal{O}},{\mathcal{D}}).$$ Similarly $$\begin{cases}
{\mathfrak{L}}({\mathcal{O}})
\subseteq
{\mathcal{O}}
\\
(\forall t>0)
({\mathcal{O}}_{t}
\bullet
{\mathcal{D}}
\subseteq
\Gamma(\pi))
\end{cases}
\Rightarrow
{\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}
\text{has the }
{\mathbf{LD}}({\mathcal{O}},{\mathcal{D}}).$$

A useful property is the following one

\[21120901\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure satisfying , $x_{\infty}\in X$. Set $S_{z}=\{B_{l}^{z}\mid l\in L\}$, then $\forall z\in X$, $\forall G\in{\mathfrak{L}}_{1}({\mathbb {R}}^{+},
{\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z});\mu_{\lambda})$ and $\forall w_{z}\in
\bigcup_{l\in L}B_{l}^{z}$ $$\label{14481001}
\left(\int_{0}^{\infty}
e^{-\lambda s}
G(s)
\,ds\right)
w_{z}
=
\int_{0}^{\infty}
e^{-\lambda s}
G(s)w_{z}\,ds.$$ Here in the second member the integration is with respect to the lct on ${\mathfrak{E}}_{z}$, while in the first member the integration is with respect to the lct on ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$.

Let $z\in X$ and $v\in\bigcup_{l\in L}B_{l}^{z}={\mathfrak{E}}_{z}$ then map ${\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z})\ni A
\mapsto
Av\in{\mathfrak{E}}_{z}$ is linear and continuous. Indeed let $l(v)\in L$ such that $v\in B_{l(v)}^{z}$, thus we have $\nu_{j}^{z}(Av)
\leq
\sup_{w\in B_{l(v)}^{z}}
\nu_{j}^{z}(Aw)
\doteq
p_{j,l(v)}^{z}
(A)$. Hence by a well-known result in vector valued integration we have .

\[21500412b\] Let ${\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}
$ be a bundle of $\Omega-$spaces and ${\mathcal{E}}\subseteq\prod_{x\in X}{\mathfrak{E}}_{x}$. Set for all $v\in\prod_{x\in X}{\mathfrak{E}}_{x}$ $$\label{11121419b}
\begin{cases}
B_{v}:X\ni x
\mapsto
\{v(x)\},
\\
\Theta
\doteqdot
\left\{
B_{w}
\mid
w\in{\mathcal{E}}
\right\}
\end{cases}$$ Thus $\Theta\subset
\prod_{x\in X}
Bounded({\mathfrak{E}}_{x})$ and $\forall v\in{\mathcal{E}}$ $$\label{01592912}
{\mathcal{E}}
\cap
B_{v}
=
\{v\}.$$ Therefore for all $v\in{\mathcal{E}}$, and for all $x\in X$ with the notations of Def. \[10282712\] $$\begin{cases}
{\mathbf{D}}(B_{v},{\mathcal{E}})
=
\{v\},
\\
{\mathcal{B}}_{B_{v}}^{x}
=\{v(x)\},\,
\\
S_{x}=\{\{w(x)\}
\mid w\in{\mathcal{E}}\},
\\
{\mathcal{E}}(\Theta)
=
{\mathcal{E}}.
\end{cases}$$

\[15210503\] Let ${\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\|\cdot\|{\right\rangle}}$ be a Banach bundle. Let $x_{\infty}\in X$ and $
{\mathcal{U}}_{0}
\in\prod_{x\in X_{0}}
{\mathcal{C}_{}
\left({\mathbb {R}}^{+},B_{s}({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x}$ for all $x\in X_{0}$. Moreover let us denote by $T_{x}$ the infinitesimal generator of the semigroup ${\mathcal{U}}_{0}(x)$ for any $x\in X_{0}$ and set $$\label{15482601}
\begin{cases}
{\mathcal{T}}_{0}(x)
\doteqdot
Graph(T_{x}),\,
x\in X_{0}
\\
\Phi
\doteqdot
\{
\phi\in\Gamma^{x_{\infty}}
(\pi_{{\mathbf{E}}^{\oplus}})
\mid
(\forall x\in X_{0})
(\phi(x)\in{\mathcal{T}}_{0}(x))
\}
\\
{\mathcal{E}}
\doteqdot
\{
v
\in
\Gamma(\pi)
\mid
(\exists\,\phi\in\Phi)
(v(x_{\infty})=\phi_{1}(x_{\infty}))
\}
\\
\Theta
\doteqdot
\left\{
B_{w}
\mid
w\in{\mathcal{E}}
\right\},
\end{cases}$$ where ${\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}^{\oplus}),\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}){\right\rangle}},\pi_{{\mathbf{E}}^{\oplus}},X,{\mathfrak{n}}^{\oplus}{\right\rangle}}$ is the bundle direct sum of the family $\{{\mathfrak{V}},
{\mathfrak{V}}\}$.

The following is a direct generalization of the definition given in [@kurtz Lm. $2.11$]

\[14531903\] Let $
{\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}
$ be a bundle of $\Omega-$spaces, where ${\mathfrak{N}}\doteqdot\{\nu_{j}\mid j\in J\}$. Moreover let $Y$ be a topological space, $s_{0}\in Y$, $f\in
\prod_{x\in X}{\mathfrak{E}}_{x}^{Y}$ and $\{z_{n}\}_{n\in{\mathbb{N}}}\subset X$. Then we say that $\{f(z_{n})\}_{n\in{\mathbb{N}}}$ is *bounded* if $\sup_{(n,s)\in{\mathbb{N}}\times Y}
\nu_{j}(f(z_{n})(s))<\infty$ for all $j\in J$. $\{f(z_{n})\}_{n\in{\mathbb{N}}}$ is *equicontinuous at $s_{0}$* if for all $j\in J$ and for all ${\varepsilon}>0$ there exists a neighbourhood $U$ of $s_{0}$ such that for all $s\in U$ we have $\sup_{n\in{\mathbb{N}}}\nu_{j}(f(z_{n})(s)
-f(z_{n})(s_{0}))\leq{\varepsilon}$. Finally $\{f(z_{n})\}_{n\in{\mathbb{N}}}$ is *equicontinuous* if $\{f(z_{n})\}_{n\in{\mathbb{N}}}$ is equicontinuous at $s$ for every $s\in Y$.

\[19492307\] Let us assume the notations of Def. \[15210503\] and that ${\mathfrak{V}}$ is full. Thus $\{v(x_{\infty})\mid v\in{\mathfrak{E}}\}
=
\{\phi_{1}(x_{\infty})\mid\phi\in\Phi\}$.

By definition follows the inclusion $\subseteq$. ${\mathfrak{V}}$ being full we have $(\forall\phi\in\Phi)
(\exists\,v\in\Gamma(\pi))
(v(x_{\infty})=\phi_{1}(x_{\infty}))$. Thus $(\forall\phi\in\Phi)
(\exists\,v\in{\mathcal{E}})
(v(x_{\infty})=\phi_{1}(x_{\infty}))$ hence the inclusion $\supseteq$.

\[17301812b\] Let ${\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\|\cdot\|{\right\rangle}}$ be a Banach bundle. Let $x_{\infty}\in X$ and $
{\mathcal{U}}_{0}
\in\prod_{x\in X_{0}}
{\mathcal{C}_{}
\left({\mathbb {R}}^{+},B_{s}({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x}$ for all $x\in X_{0}$.

$D(T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$ and $\exists\lambda_{0}>0$ (respectively $\exists\lambda_{0}>0,
\lambda_{1}<0$) such that the range ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$, (respectively the ranges ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ and ${\mathcal{R}}(\lambda_{1}-T_{x_{\infty}})$ are dense in ${\mathfrak{E}}_{x_{\infty}}$), $${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}
\in
Graph\left({\mathfrak{V}},{\mathfrak{V}}\right),$$ and $T_{x_{\infty}}$ in is the generator of a $C_{0}-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x_{\infty}}$.

Moreover that $\{v(x)\mid v\in{\mathcal{E}}\}$ is dense in ${\mathfrak{E}}_{x}$ for all $x\in X_{0}$, by taking the notations in , let ${\mathfrak{W}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ and ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure [^14] such that holds. Assume $
{\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{z})}}
({\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z}))
\subseteq
{\mathfrak{M}}_{z}$ (respectively ${\mathbf{U}}_{is}
({\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z}))
\subseteq
{\mathfrak{M}}_{z}$) for all $z\in X$ [^15] and that there exists $F\in\Gamma(\rho)$ such that $F(x_{\infty})={\mathcal{U}}(x_{\infty})$ and

i

:   ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the ${\mathbf{LD}}_{x_{\infty}}(\{F\},{\mathcal{E}})$; **or** it has the ${\mathbf{LD}}(\{F\},{\mathcal{E}})$;

ii

:   $(\forall v\in{\mathcal{E}})
    (\exists\,\phi\in\Phi)$ s.t. $\phi_{1}(x_{\infty})
    =
    v(x_{\infty})$ and $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X
    \mid
    \lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ we have that $\{
    {\mathcal{U}}(z_{n})(\cdot)\phi_{1}(z_{n})
    -
    F(z_{n})(\cdot)v(z_{n})
    \}_{n\in{\mathbb{N}}}$ is a bounded equicontinuous sequence;

iii

:   $X$ is metrizable.

$(\forall v\in{\mathcal{E}})
(\forall K\in Compact({\mathbb {R}}^{+}))$ $$\label{02502912}
\boxed{
\lim_{z\to x_{\infty}}
\sup_{s\in K}
\left\|
{\mathcal{U}}(z)(s)v(z)
-
F(z)(s)v(z)
\right\|=0,}$$ and $$\label{02512912p}
\boxed{
{\mathcal{U}}
\in\Gamma^{x_{\infty}}(\rho)
.}$$ In particular $$\label{02512912}
\{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}\}
\in
\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}.$$ Here ${\mathcal{T}}$ and $D(T_{x_{\infty}})$ are defined as in Notations \[15411512b\] with ${\mathcal{T}}_{0}$ and $\Phi$ given in , while ${\mathcal{U}}
\in\prod_{x\in X}
{\mathfrak{M}}_{x}$ such that ${\mathcal{U}}{\upharpoonright}X_{0}\doteqdot
{\mathcal{U}}_{0}$ and ${\mathcal{U}}(x_{\infty})$ is the semigroup on ${\mathfrak{E}}_{x_{\infty}}$ generated by $T_{x_{\infty}}$.

By Lemma \[13001512b\], [@kurtz Lms.$(2.8)-(2.9)$], and the Hille-Yosida theorem, see [@kurtz Th.$(1.2)$], we have the first sentence of the statement for the case of semigroup of contractions. By [@bra Corollary $3.1.19.$] applied to $T_{x}$, for any $x\in X_{0}$, and by we have $(\forall\lambda\in{\mathbb {R}})
(\forall v_{x_{\infty}}\in 
Dom(T_{x_{\infty}}))$ $$\label{15363001}
\|({\mathbf{1}}-\lambda T_{x_{\infty}})
v_{x_{\infty}})\|_{x_{\infty}}
\geq
\|v_{x_{\infty}}\|_{x_{\infty}}.$$ Hence by [@bra Corollary $3.1.19.$], $T_{x_{\infty}}$ will be a generator of a strongly continuous semigroup of isometries if we show that $\forall\lambda\in{\mathbb {R}}-\{0\}$ $$\label{15443001}
{\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}})
=
{\mathfrak{E}}_{x_{\infty}}.$$ Let us set $$\rho_{0}(T_{x_{\infty}})
\doteqdot
\{
\lambda\in
{\mathbb {R}}-\{0\}
\mid
{\mathcal{R}}({\mathbf{1}}-\lambda T)
=
{\mathfrak{E}}_{x}\}.$$ By $\rho_{0}(T_{x_{\infty}})
=
\rho(T_{x_{\infty}})
\cap
({\mathbb {R}}-\{0\})$, where $\rho(T_{x_{\infty}})$ is the resolvent set of $T_{x}$. By [@ds Lemma $7.3.2$] $\rho(T_{x_{\infty}})$ is open in ${\mathbb {C}}$ so $\rho_{0}(T_{x_{\infty}})$ is open in ${\mathbb {R}}-\{0\}$ with respect to the topology on ${\mathbb {R}}-\{0\}$ induced by that on ${\mathbb {C}}$. By Lemma \[14253001\] we deduce that $\rho_{0}(T_{x_{\infty}})$ is also closed in ${\mathbb {R}}-\{0\}$, therefore $\rho_{0}(T_{x_{\infty}})={\mathbb {R}}-\{0\}$ and follows as well that $T_{x_{\infty}}$ is a generator of a strongly continuous semigroup of isometries.

Now we shall apply Lemma \[15482712\] in order to show the remaining part of the statement. By the Dupre’ Thm., see for example [@kurtz Cor. $2.10$], and the fact that a metrizable space is completely regular, we deduce by hyp. $(iii)$ that ${\mathfrak{V}}$ is full. Let $v\in{\mathcal{E}}$ be fixed then by , $(\exists\,\phi\in\Phi)
(v(x_{\infty})=\phi_{1}(x_{\infty}))$ thus by and Corollary \[28111707\] $$\label{20352312b}
\lim_{z\to x_{\infty}}
\|v(z)-\phi_{1}(z)\|
=0.$$ Now let $F\in\Gamma(\rho)$ of which in hypothesis so in particular $$\label{10182312b}
F(x_{\infty})={\mathcal{U}}(x_{\infty}),$$ moreover $\forall s\in{\mathbb {R}}^{+}$ and $z\in X$ $$\begin{aligned}
{2}
\label{20492312b}
\|
{\mathcal{U}}(z)(s)v(z)
-
F(z)(s)v(z)
\|
&
\leq
\notag
\\
\|
{\mathcal{U}}(z)(s)v(z)
-
{\mathcal{U}}(z)(s)\phi_{1}(z)
\|
+
\|{\mathcal{U}}(z)(s)\phi_{1}(z)-F(z)(s)v(z)\|
&
\leq
\notag
\\
\|v(z)-\phi_{1}(z)\|
+
\|{\mathcal{U}}(z)(s)\phi_{1}(z)-F(z)(s)v(z)\|.\end{aligned}$$ For any $\lambda>0$ let us set $$g_{\infty}^{\lambda}
\doteqdot
(\lambda-T_{x_{\infty}})^{-1}
\phi_{1}(x_{\infty})$$ thus $g_{\infty}^{\lambda}
\in Dom(T_{x_{\infty}})$ hence by Remark \[18581512b\] and the construction of $T_{x_{\infty}}$ $
\exists\,
\psi^{\lambda}
\in
\Phi
$ such that $$\label{09252412b}
\begin{cases}
g_{\infty}^{\lambda}
=
\psi_{1}^{\lambda}(x_{\infty})
=
\lim_{z\in x_{\infty}}
\psi_{1}^{\lambda}(z)
\\
T_{x_{\infty}}
g_{\infty}^{\lambda}
=
\lim_{z\to x_{\infty}}
T_{z}
\psi_{1}^{\lambda}(z).
\end{cases}$$ By and for all $z\in X$ and for all $w_{z}\in
\bigcup_{v\in{\mathcal{E}}}v(z)$ $$\label{15261001pre}
\left(\int_{0}^{\infty}
e^{-\lambda s}
F(z)(s)
\,ds\right)
w_{z}
=
\int_{0}^{\infty}
e^{-\lambda s}
F(z)(s)w_{z}\,ds.$$ Moreover by the fact that ${\mathfrak{V}}$ is full we have that for all $\phi\in\Phi$ there exists a $v\in\Gamma(\pi)$ such that $v(x_{\infty})=\phi_{1}(x_{\infty})$, thus by construction of ${\mathcal{E}}$ $$\label{18390103}
(\forall\phi\in\Phi)
(\exists\,v\in{\mathcal{E}})
(v(x_{\infty})=\phi_{1}(x_{\infty})).$$ Hence by , and for all $\phi\in\Phi$ $$\label{15261001}
\left(\int_{0}^{\infty}
e^{-\lambda s}
F(x_{\infty})(s)
\,ds\right)
\phi_{1}(x_{\infty})
=
\int_{0}^{\infty}
e^{-\lambda s}
{\mathcal{U}}(x_{\infty})(s)
\phi_{1}(x_{\infty})
\,ds.$$ Now set $$\xi
\doteqdot
{\mathfrak{L}}(F),$$ thus by hypothesis $(i)$ we have for all $\lambda>0$ $$\label{11462412b}
\xi(\cdot)(\lambda)v(\cdot)
\in
\Gamma^{x_{\infty}}(\pi).$$ Moreover $$\begin{aligned}
{2}
\label{10262412b}
\xi(x_{\infty})(\lambda)
v(x_{\infty})
&
=
\xi(x_{\infty})(\lambda)
\phi_{1}(x_{\infty})
\notag
\\
&
=
\int_{0}^{\infty}
e^{-\lambda s}
{\mathcal{U}}(x_{\infty})(s)
\phi_{1}(x_{\infty})
\,ds\,
\text{ by }
\eqref{15261001}
\notag
\\
&
=
(\lambda-T_{x_{\infty}})^{-1}
\phi_{1}(x_{\infty})\,
\text{ by \cite[$(1.3)$]{kurtz}}
\notag
\\
&
\doteq
g_{\infty}^{\lambda}
=
\psi_{1}^{\lambda}(x_{\infty})
\,
\text{ by } 
\eqref{09252412b}.\end{aligned}$$ By the fact that ${\mathfrak{V}}$ is full, by , the fact that $\psi_{1}^{\lambda}
\in\Gamma^{x_{\infty}}(\pi)$ by , by and by Corollary \[21492812\] we have $\forall\lambda>0$ $$\label{10352412b}
\lim_{z\to x_{\infty}}
\|\psi_{1}^{\lambda}(z)-
\xi(z)(\lambda)v(z))\|
=0.$$ Now $(\forall\lambda>0)(\forall z\in X)$ set $$w^{\lambda}(z)
\doteqdot
(\lambda{\mathbf{1}}-T_{z})
\psi_{1}^{\lambda}(z),$$ thus $$\begin{aligned}
{2}
\label{12042412b}
\left\|
\int_{0}^{\infty}
e^{-\lambda s}
\left(
{\mathcal{U}}(z)(s)\phi_{1}(z)-F(z)(s)v(z)
\right)
\,ds
\right\|
&
\leq
\notag
\\
\left\|
\int_{0}^{\infty}
e^{-\lambda s}
{\mathcal{U}}(z)(s)(\phi_{1}(z)-w^{\lambda}(z))
\,ds
\right\|
+
\left\|
\int_{0}^{\infty}
e^{-\lambda s}
\left(
{\mathcal{U}}(z)(s)w^{\lambda}(z)-F(z)(s)v(z)
\right)
\,ds
\right\|
&
\leq
\notag
\\
\frac{1}{\lambda}
\|\phi_{1}(z)-w^{\lambda}(z)\|
+
\|\psi_{1}^{\lambda}(z)-
\xi(z)(\lambda)v(z))
&\|.\end{aligned}$$ Here we consider that by hypothesis and by the first part of the statemet $\|{\mathcal{U}}(z)\|\leq 1$ for all $z\in X$, moreover we applied the Hille-Yosida formula [@kurtz $(1.3)$]. Now $$\begin{aligned}
{1}
\label{11272412b}
\|\phi_{1}(z)-w^{\lambda}(z)\|
&
=
\\
\|
\phi_{1}(z)-
(\lambda{\mathbf{1}}-T_{z})
\psi_{1}^{\lambda}(z)
\|
&
\leq
\notag
\\
\|
\phi_{1}(z)-v(z)
\|
+
\|v(z)-\lambda\xi(z)(\lambda)v(z)
+
\lambda\xi(z)(\lambda)v(z)
-(\lambda{\mathbf{1}}-T_{z})
\psi_{1}^{\lambda}(z)
\|
&
\leq
\notag
\\
\|\phi_{1}(z)-v(z)\|
+
\lambda
\|\xi(z)(\lambda)v(z)-\psi_{1}^{\lambda}(z)\|
+
\|
T_{z}\psi_{1}^{\lambda}(z)
-(\lambda\xi(z)(\lambda)v(z)-v(z))
&
\|.
\notag\end{aligned}$$ By $
T_{x_{\infty}}
\psi_{1}^{\lambda}(x_{\infty})
=
T_{x_{\infty}}
g_{\infty}^{\lambda}
$ moreover $$\begin{aligned}
{1}
\label{11402412b}
T_{x_{\infty}}
g_{\infty}^{\lambda}
&
=
-(\lambda-T_{x_{\infty}})
g_{\infty}^{\lambda}
+\lambda g_{\infty}^{\lambda}
\notag
\\
&
=
-(\lambda-T_{x_{\infty}})
(\lambda-T_{x_{\infty}})^{-1}
\phi_{1}(x_{\infty})
+\lambda g_{\infty}^{\lambda}
\notag
\\
&
=
\lambda g_{\infty}^{\lambda}
-
\phi_{1}(x_{\infty})
=
\lambda\xi(x_{\infty})(\lambda)
v(x_{\infty})
-
v(x_{\infty}),\end{aligned}$$ where in the last equality we used and the construction of $\phi$. By we have that $(X\ni z
\mapsto
T_{z}
\psi_{1}^{\lambda}(z))
\in
\Gamma^{x_{\infty}}(\pi)
$, hence by , the fact that $\lambda\xi(\cdot)(\lambda)v(\cdot)
-
v\in\Gamma^{x_{\infty}}(\pi)
$ by , we deduce by the fact that ${\mathfrak{V}}$ is full and by Corollary \[21492812\] that $\forall\lambda>0$ $$\label{11542412b}
\lim_{z\to x_{\infty}}
\|T_{z}\psi_{1}^{\lambda}(z)
-(\lambda\xi(z)(\lambda)v(z)-v(z))\|=0.$$ Therefore by , , and $$\lim_{z\to x_{\infty}}
\|\phi_{1}(z)-w^{\lambda}(z)\|
=0.$$ By this one along with we can state by using that $\forall\lambda>0$ $$\lim_{z\to x_{\infty}}
\left\|
\int_{0}^{\infty}
e^{-\lambda s}
\left(
{\mathcal{U}}(z)(s)\phi_{1}(z)-F(z)(s)v(z)
\right)
\,ds
\right\|
=0.$$ Therefore $\forall\lambda>0$ and $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X
\mid
\lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ $$\label{12052412b}
\lim_{n\in{\mathbb{N}}}
\left\|
\int_{0}^{\infty}
e^{-\lambda s}
\left(
{\mathcal{U}}(z_{n})(s)\phi_{1}(z_{n})-
F(z_{n})(s)v(z_{n})
\right)
\,ds
\right\|
=0.$$ By , hypothesis $(ii)$ and [@kurtz Lemma $(2.11)$] we have $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X
\mid
\lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ and $\forall K\in Compact({\mathbb {R}}^{+})$ $$\lim_{n\in{\mathbb{N}}}
\sup_{s\in K}
\left\|
{\mathcal{U}}(z_{n})(s)\phi_{1}(z_{n})
-
F(z)(s)v(z_{n})
\right\|=0.$$ Therefore by hypothesis $(iii)$ and [@BourGT Prop.$10$, $\S2.6$, Ch. $9$], $\forall K\in Compact({\mathbb {R}}^{+})$ $$\label{12102412b}
\lim_{z\to x_{\infty}}
\sup_{s\in K}
\left\|
{\mathcal{U}}(z)(s)\phi_{1}(z)
-
F(z)(s)v(z)
\right\|=0.$$ In conclusion by , and we obtain $\forall K\in Compact({\mathbb {R}}^{+})$ $$\label{02012912}
\lim_{z\to x_{\infty}}
\sup_{s\in K}
\left\|
{\mathcal{U}}(z)(s)v(z)
-
F(z)(s)v(z)
\right\|=0,$$ hence . By and we obtain . Thus and follow by Lemma \[15482712\], by and by the following one $\forall K\in Compact({\mathbb {R}}^{+})$ and $\forall v\in{\mathcal{E}}$ $$\sup_{z\in X}
\sup_{s\in K}
\left\|
{\mathcal{U}}(z)(s)v(z)
\right\|
\leq
\sup_{z\in X}
\|v(z)\|
<\infty.$$ where we considered that by construction $\left\|
{\mathcal{U}}(z)(s)
\right\|\leq
1$, for all $s\in{\mathbb {R}}^{+}$ and $z\in X$ and that $v\in\Gamma(\pi)$.

\[17341602\] If ${\mathfrak{W}}$ is full $(\exists\,F\in\Gamma(\rho))
(F(x_{\infty})={\mathcal{U}}(x_{\infty}))$, so hypotheses reduce.

Corollary I.Constructions of Equicontinuous sequence
----------------------------------------------------

As the first corollary we give conditions in order to satisfy the bounded equicontinuity of which in hypothesis $(ii)$.

\[21343012\] Let us assume the hypotheses of Theorem \[17301812b\] except $(ii)$ replaced by the following one $$(\exists\,G\in\prod_{z\in X}
{\mathfrak{L}}_{1}\left({\mathbb {R}}^{+},
{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)
(\exists\,H\in
\prod_{z\in X}^{b}{\mathcal{L}}({\mathfrak{E}}_{z}))
(\exists\,F\in\Gamma(\rho))$$ such that $F(x_{\infty})={\mathcal{U}}(x_{\infty})$ and $\forall s>0$ $$\label{22053012}
\begin{cases}
\sup_{x\in X}\sup_{s>0}\|F(x)(s)\|<\infty
\\
(\forall s_{1}>0)
(\exists\, a>0)
(\sup_{u\in[s_{1},s]}
\sup_{z\in X}
\|G(z)(u)\|\leq a |s-s_{1}|)
\\
(\forall z\in X)
(F(z)(s)
=
H(z)+
\int_{0}^{s}
G(z)(u)\,du),
\end{cases}$$ where the integration is with respect to the Lebesgue measure on $[0,s]$ and with respect to the $lct$ on ${\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z})$. Then holds the statement of Theorem \[17301812b\].

Let $v\in{\mathcal{E}}$ thus $(\exists\,\phi\in\Phi)
(v(x_{\infty})=\phi_{1}(x_{\infty}))$ so $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X
\mid
\lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ we have $$\sup_{n\in{\mathbb{N}}}
\sup_{s>0}
\|{\mathcal{U}}(z_{n})(s)
\phi_{1}(z_{n})
-
F(z_{n})(s)v(z_{n})
\|
\leq
\sup_{n\in{\mathbb{N}}}
\|\phi_{1}(z_{n})\|
+
M
\sup_{n\in{\mathbb{N}}}
\|v(z_{n})\|
<\infty.$$ Here in the first inequality we used $\|{\mathcal{U}}(z)(s)\|\leq 1$ for all $z\in X$ and $s>0$ by construction, and $M\doteqdot
\sup_{z\in X}\sup_{s>0}\|F(z)(s)\|<\infty$ by hypothesis, while in the second inequality we used the fact that $v\in\prod_{x\in X}^{b}{\mathfrak{E}}_{x}$, by construction and that $\sup_{n\in{\mathbb{N}}}
\|\phi_{1}(z_{n})\|<\infty$ because of $\exists\,
\varlimsup_{n\in{\mathbb{N}}}\|\phi_{1}(z_{n})\|
\in{\mathbb {R}}$ by Remark \[18581512b\] and by construction $\|\cdot\|$ is $u.s.c.$ Moreover by [@kurtz $(1.4)$], and $S_{x}=
\{\{w(x)\}\mid w\in{\mathcal{E}}\}$ for all $x\in X$ we have $$\begin{aligned}
{1}
{\mathcal{U}}(z_{n})(s)
\phi_{1}(z_{n})
-
F(z_{n})(s)v(z_{n})
&
=
\int_{0}^{s}
\left(
{\mathcal{U}}(z_{n})(u)
T_{z_{n}}
\phi_{1}(z_{n})
-
G(z_{n})(u)v(z_{n})
\right)
\,du
+
\\
&
+
\phi_{1}(z_{n})
-
H(z_{n})v(z_{n}).\end{aligned}$$ Thus for any $s_{1},s_{2}\in{\mathbb {R}}^{+}$ $$\begin{aligned}
{1}
\sup_{n\in{\mathbb{N}}}
\|
({\mathcal{U}}(z_{n})(s_{1})
\phi_{1}(z_{n})
-
F(z_{n})(s_{1})v(z_{n}))
-
({\mathcal{U}}(z_{n})(s_{2})
\phi_{1}(z_{n})
-
F(z_{n})(s_{2})v(z_{n}))
\|
&
\leq
\\
|s_{1}-s_{2}|
\sup_{n\in{\mathbb{N}}}
\sup_{u\in[s_{1},s_{2}]}
\|{\mathcal{U}}(z_{n})(u)
T_{z_{n}}
\phi_{1}(z_{n})
-
G(z_{n})(u)v(z_{n})
\|
&
\leq
\\
|s_{1}-s_{2}|
\sup_{n\in{\mathbb{N}}}
(\|T_{z_{n}}\phi_{1}(z_{n})\|
-a\|v(z_{n})\|)
&
\leq
J|s_{1}-s_{2}|.\end{aligned}$$ Here in the second inequality we used $\|{\mathcal{U}}(z)(u)\|\leq 1$ by construction and the hypothesis, in the third one the fact that $\sup_{n\in{\mathbb{N}}}
\|T_{z_{n}}\phi_{1}(z_{n})\|
<\infty$ as well $\sup_{n\in{\mathbb{N}}}\|v(z_{n})\|<\infty$, because of $\exists\,
\varlimsup_{n\in{\mathbb{N}}}
\|T_{z_{n}}\phi_{1}(z_{n})\|
\in{\mathbb {R}}$ and $
\exists\,
\varlimsup_{n\in{\mathbb{N}}}\|v(z_{n})\|
\in{\mathbb {R}}$ due to the fact that $\|\cdot\|$ is $u.s.c.$ by construction and Remark \[18581512b\] for the first limit and the continuity of $v$ for the second one. Therefore hypothesis $(ii)$ of Theorem \[17301812b\] is satisfied, hence the statement follows by Theorem \[17301812b\].

Corollaries II. Construction of ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ with the ${\mathbf{LD}}$  {#13320803}
-------------------------------------------------------------------------------------------------------------------------------------------

Let us start with the following simple result about the relation among full and Laplace duality property.

\[19030106\] Let ${\mathfrak{W}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ and ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that ${\mathfrak{V}}$ is a Banach bundle and $x_{\infty}\in X$. Assume that

1.  ${\mathfrak{V}}$ and ${\mathfrak{W}}$ are full;

2.  ${\mathcal{E}}=\Gamma(\pi)$ and $\Theta$ is given in ;

3.  $(\forall F\in\Gamma^{x_{\infty}}(\rho))
    (M(F)\doteqdot
    \sup_{x\in X}
    \sup_{s\in{\mathbb {R}}^{+}}
    \|F(x)(s)\|<\infty)$;

4.  $(\forall\sigma\in\Gamma(\rho))
    (\sup_{x\in X}
    \sup_{s\in{\mathbb {R}}^{+}}
    \|\sigma(x)(s)\|<\infty)$;

5.  $X$ is metrizable.

If ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the ${\mathbf{LD}}$ then it has the ${\mathbf{LD}}_{x_{\infty}}$.

Let $F\in\Gamma^{x_{\infty}}(\rho)$ and $w\in\Gamma^{x_{\infty}}(\pi)$ thus by hypothesis $(2)$ and Corollary \[28111707\] there exist $\sigma\in\Gamma(\rho)$ and $v\in\Gamma(\pi)$ such that $\sigma(x_{\infty})=F(x_{\infty})$, $v(x_{\infty})=w(x_{\infty})$, and $\forall K\in Comp({\mathbb {R}}^{+})$, $\forall v\in{\mathcal{E}}$ $$\label{19330106}
\begin{cases}
\lim_{z\to x_{\infty}}
\|w(z)-v(z)\|=0
\\
\lim_{z\to x_{\infty}}
\sup_{s\in K}
\|(F(x)(s)-\sigma(x)(s))v(x)\|
&=0.
\end{cases}$$ Moreover $\forall\lambda>0$ $$\begin{aligned}
{1}
\label{20090106}
\left\|
\int_{0}^{\infty}
e^{\lambda s}
F(z)(s)w(z)\,ds
-
\int_{0}^{\infty}
e^{\lambda s}
\sigma(z)(s)v(z)\,ds
\right\|
&\leq
\notag
\\
\left\|
\int_{0}^{\infty}
e^{\lambda s}
F(z)(s)(w(z)-v(z))\,ds
\right\|
+
\left\|
\int_{0}^{\infty}
e^{\lambda s}
(F(z)(s)-\sigma(z)(s))v(z)\,ds
\right\|
&\leq
\notag
\\
\frac{1}{\lambda}M(F)\|v(z)-w(z)\|
+
\int_{0}^{\infty}
e^{\lambda s}
\left\|
(F(z)(s)-\sigma(z)(s))v(z)
\right\|\,ds.\end{aligned}$$ By the hypotheses $(3-4)$ $\sup_{z\in X}
\sup_{s\in{\mathbb {R}}^{+}}
\left\|(F(z)(s)-\sigma(z)(s))v(z)\right\|
<\infty$ hence $\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X$ such that $\lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty}$ we have by , and a well-known theorem on convergence of sequences of integrals that $\forall\lambda>0$ $$\lim_{n\in{\mathbb{N}}}
\left\|
\int_{0}^{\infty}
e^{\lambda s}
F(z_{n})(s)w(z_{n})\,ds
-
\int_{0}^{\infty}
e^{\lambda s}
\sigma(z_{n})(s)v(z_{n})\,ds
\right\|
=0.$$ Thus $\forall\lambda>0$ by hypothesis $(5)$ $$\lim_{z\to x_{\infty}}
\left\|
\int_{0}^{\infty}
e^{\lambda s}
F(z)(s)w(z)\,ds
-
\int_{0}^{\infty}
e^{\lambda s}
\sigma(z)(s)v(z)\,ds
\right\|
=0,$$ hence the statement by Corollary \[28111707\].

Now we shall see that in the case of a bundle of normed space we can choose for all $x$ a simple space ${\mathfrak{M}}_{x}$ satisfying .

\[18390901\] Let ${\mathfrak{W}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ and ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that for all $x\in X$, ${\mathfrak{E}}_{x}$ is a reflexive Banach space, $S_{x}\subseteq{\mathcal{P}}_{\omega}({\mathfrak{E}}_{x})$ and $${\mathfrak{M}}_{x}
\subseteq
\left\{
F\in
{\mathcal{C}_{c}
\left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)}
\mid
(\forall\lambda>0)
\left(\int_{{\mathbb {R}}^{+}}^{*}
e^{-\lambda s}
\|F(s)\|_{B({\mathfrak{E}}_{x})}
\,ds
<\infty\right)
\right\}.$$ Thus $$\label{14441001}
{\mathfrak{M}}_{x}
\subset
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}
({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});
\mu_{\lambda}).$$ In particular , and ${\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{x})}}
({\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}))
\subseteq
{\mathfrak{M}}_{x}$ hold if for any $x\in X$ $${\mathfrak{M}}_{x}
=
\left\{
F\in
{\mathcal{C}_{c}
\left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)}
\mid
\sup_{s\in{\mathbb {R}}^{+}}
\|F(s)\|_{B({\mathfrak{E}}_{x})}
<\infty
\right\}.$$

The first sentence follows by [@SilInt Corollary $2.6.$], while the second sentence comes by the first one.

### ${\mathbf{U}}-$Spaces

\[12042101\] First of all recall that for any $W,Z$ topological vector spaces over ${\mathbb{K}}\in\{{\mathbb {R}},{\mathbb {C}}\}$ we denote by ${\mathcal{L}}(W,Z)$ the ${\mathbb{K}}-$linear space of all continuous linear map on $W$ and with values in $Z$ and set ${\mathcal{L}}(Z)
\doteqdot
{\mathcal{L}}(Z,Z)$ and $Z^{*}
\doteqdot
{\mathcal{L}}(Z,{\mathbb{K}})$. In this section we assume fixed the following data:

1.  a set $X$, a locally compact space $Y$ and a Radon measure $\mu$ on $Y$;

2.  a family $\{{\mathfrak{E}}_{x}\}_{x\in X}$ of $Hlcs$;

3.  a family $\{\tau_{x}\}_{x\in X}$ such that ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}
    \in Hlcs$, $\forall x\in X$;

4.  a family $\{{\mathfrak{N}}_{x}\}_{x\in X}$ such that ${\mathfrak{N}}_{x}\doteqdot\{\nu_{j_{x}}^{x}
    \mid j_{x}\in J_{x}\}$ is a fundamental set of seminorms on ${\mathfrak{E}}_{x}$, $\forall x\in X$;

5.  a family $\{Q_{x}\}_{x\in X}$ such that $Q_{x}\doteqdot\{q_{\alpha_{x}}^{x}
    \mid \alpha_{x}\in A_{x}\}$ is a fundamental set of seminorms on ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}$, $\forall x\in X$;

6.  ${\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}\in Hlcs$ such that

    -   ${\mathcal{H}}
        \subseteq
        \prod_{x\in X}{\mathfrak{E}}_{x}$ as linear spaces;

    -   $\imath_{x}({\mathfrak{E}}_{x})
        \subset{\mathcal{H}}$, for all $x\in X$;

    -   $\Pr_{x}\in
        {\mathcal{L}}\left(
        {\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}},
        {\mathfrak{E}}_{x}
        \right)$ and $\imath_{x}\in
        {\mathcal{L}}\left({\mathfrak{E}}_{x},
        {\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}\right)$, for all $x\in X$;

    -   $\exists\,{\mathcal{A}}\subseteq
        \prod_{x\in X}{\mathcal{L}}({\mathfrak{E}}_{x})$ linear space such that

        1.  $\theta({\mathcal{A}}){\upharpoonright}{\mathcal{H}}
            \subseteq
            {\mathcal{L}}({\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}})$,

        2.  $\imath_{x}
            ({\mathcal{L}}({\mathfrak{E}}_{x}))
            \subseteq{\mathcal{A}}$ for all $x\in X$.

Here $\theta$ is defined in Def. \[13021401\].

For any $Z\in Hlcs$ we denote by ${\mathfrak{L}}_{1}(Y,Z,\mu)$ the linear space of all maps on $Y$ and with values in $Z$ which are essentially $\mu-$integrable in the sense described in [@IntBourb Ch. $6$]. While ${\mathcal{C}_{cs}
\left(Y,Z\right)}$ denotes the linear space of all continuous maps $f:Y\to Z$ with compact support. Moreover for any family $\{Z_{x}\}_{x\in X}$ of linear spaces and for all $x\in X$ set $\Pr_{x}:
\prod_{y\in X}Z_{y}
\ni
f\mapsto f(x) 
\in
Z_{x}$ and $\imath_{x}:
Z_{x}
\to
\prod_{y\in X}Z_{y}
$ such that for all $x\neq y$ and $z_{x}\in Z_{x}$ $\Pr_{y}\circ\imath_{x}(z_{x})={\mathbf{0}}_{y}$, while $\Pr_{x}\circ\imath_{x}=Id_{x}$.

Finally we set $${\left\langle \cdot,\cdot{\right\rangle}}:
End({\mathcal{H}})
\times
{\mathcal{H}}
\ni
(A,v)
\mapsto
A(v)
\in
{\mathcal{H}},$$ and for all $x\in X$ $${\left\langle \cdot,\cdot{\right\rangle}}_{x}:
End\left({\mathfrak{E}}_{x}\right)
\times
{\mathfrak{E}}_{x}
\ni
(A,v)
\mapsto
A(v)\in
{\mathfrak{E}}_{x}.$$

\[12182503\] Let ${\mathfrak{W}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ be a bundle of $\Omega-$spaces such that for all $x\in X$ $${\mathfrak{M}}_{x}
\subseteq
{\mathfrak{L}}_{1}(Y,
{\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}};\mu).$$ Set $$\label{14162503}
\begin{cases}
\blacksquare_{\mu}:
\prod_{x\in X}
{\mathfrak{L}}_{1}(Y,
{\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}};\mu)
\times
\prod_{x\in X}
{\mathfrak{E}}_{x}
\to
\prod_{x\in X}
{\mathfrak{E}}_{x}
\\
\blacksquare_{\mu}
(H,v)(x)
\doteqdot
{\left\langle \int_{{\mathbb {R}}^{+}}
H(x)(s)\,
d\mu(s),v(x){\right\rangle}}_{x}
\in
{\mathfrak{E}}_{x}.
\end{cases}$$

\[14092503\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure satisfying and ${\mathcal{O}}\subseteq\Gamma(\rho)$, ${\mathcal{D}}\subseteq\Gamma(\pi)$. Then $$\label{LD}
{\mathbf{LD}}({\mathcal{O}},{\mathcal{D}})
\Leftrightarrow
(\forall\lambda>0)
\left(
\blacksquare_{\mu_{\lambda}}
\left({\mathcal{O}},{\mathcal{D}}\right)
\subseteq
\Gamma(\pi)
\right).$$ Similarly for all $x\in X$ $$\label{LDx}
{\mathbf{LD}}_{x}({\mathcal{O}},{\mathcal{D}})
\Leftrightarrow
(\forall\lambda>0)
\left(
\blacksquare_{\mu_{\lambda}}
\left(\Gamma_{{\mathcal{O}}}^{x}(\rho)
,\Gamma_{{\mathcal{D}}}^{x}(\pi)
\right)
\subseteq
\Gamma^{x}(\pi)
\right).$$

\[ ${\mathbf{U}}-$Spaces \] \[14302503\] ${\mathfrak{G}}$ is a ${\mathbf{U}}-$space with respect to $\{{\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}\}_{x\in X}$, ${\mathfrak{T}}$ and $D$ iff

1.  ${\mathfrak{G}}\in Hlcs$;

2.  ${\mathfrak{G}}
    \subset
    {\mathcal{L}}\left({\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}\right)$ as linear spaces;

3.  $D\subseteq{\mathcal{H}}$;

4.  $(\forall T\in lcp)
    \left(\exists\,
    \Psi_{T}\in
    End\left[
    End({\mathcal{H}})^{T},
    \prod_{x\in X}
    End({\mathfrak{E}}_{x})^{Y}
    \right]\right)
    (\forall\nu\in Radon(T))$ $$\Psi_{T}:
    {\mathfrak{L}}_{1}(T,{\mathfrak{G}},\nu)
    \to
    \prod_{x\in X}
    {\mathfrak{L}}_{1}\left(T,
    {\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}};\nu\right)
    \footnote{
    Of course 
    $\Psi_{T}$
    here
    has to be understood as
    $\Psi_{T}{\upharpoonright}{\mathfrak{L}}_{1}(T,{\mathfrak{G}},\nu)$.}
    $$ and $\forall{\overline}{F}
    \in
    {\mathfrak{L}}_{1}(T,{\mathfrak{G}},\nu)
    $, $\forall v\in D$. $\forall x\in X$ $$\label{17432203}
    \boxed{
    {\left\langle \int
    \Psi_{T}({\overline}{F})(x)(s)
    \,d\nu(s),v(x){\right\rangle}}_{x}
    =
    {\left\langle \int{\overline}{F}(s)\,d\nu(s),v{\right\rangle}}(x)
    }$$

The reason of introducing the concept of ${\mathbf{U}}-$spaces will be clarified by the following

\[14492503\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure satisfying , and let ${\mathfrak{G}}$ be a ${\mathbf{U}}-$space with respect to $\{{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\}_{x\in X}$, ${\mathfrak{T}}$ and ${\mathcal{D}}$. Then $\forall\lambda>0$, ${\overline}{F}
\in
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$, $v\in{\mathcal{D}}$ $$\label{15102503}
\blacksquare_{\mu_{\lambda}}
(\Psi_{{\mathbb {R}}^{+}}({\overline}{F}),v)
=
{\left\langle \int{\overline}{F}(s)\,d\mu_{\lambda}(s),v{\right\rangle}}.$$ Moreover if $\exists\,{\mathcal{F}}\subset
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi_{{\mathbb {R}}^{+}}({\mathcal{F}})={\mathcal{O}}$ then $$\label{21002103}
\boxed{
{\mathbf{LD}}({\mathcal{O}},{\mathcal{D}})
\Leftrightarrow
(\forall\lambda>0)
({\left\langle {\mathcal{B}}_{\lambda},{\mathcal{D}}{\right\rangle}}
\subseteq\Gamma(\pi)).}$$ Here $${\mathcal{B}}_{\lambda}
\doteqdot
\left\{
\int{\overline}{F}(s)\,d\mu_{\lambda}(s)
\mid{\overline}{F}\in{\mathcal{F}}
\right\}.$$

\[13390104\] In particular if $\exists\,{\mathcal{F}}\subset
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi_{{\mathbb {R}}^{+}}({\mathcal{F}})={\mathcal{O}}$ then $${\left\langle {\mathfrak{G}},{\mathcal{D}}{\right\rangle}}
\subseteq
\Gamma(\pi)
\Rightarrow
{\mathbf{LD}}({\mathcal{O}},{\mathcal{D}}).$$ More in general if $\exists\,{\mathfrak{G}}_{0}$ complete subspace of ${\mathfrak{G}}$ and $\exists\,{\mathcal{F}}\subset
\left\{
{\overline}{F}\in
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})
\mid
{\overline}{F}({\mathbb {R}}^{+})\subseteq{\mathfrak{G}}_{0}
\right\}$ such that $\Psi_{{\mathbb {R}}^{+}}({\mathcal{F}})={\mathcal{O}}$ then $${\left\langle {\mathfrak{G}}_{0},{\mathcal{D}}{\right\rangle}}
\subseteq
\Gamma(\pi)
\Rightarrow
{\mathbf{LD}}({\mathcal{O}},{\mathcal{D}}).$$

follows by , while follows by and Remark \[14092503\].

Thus the ${\mathbf{U}}$ property expressed by is an important tool for ensuring the satisfaction of the ${\mathbf{LD}}$. For this reason the remaining of the present section will be dedicated to the construction of a space ${\mathfrak{G}}$, Def. , which is a ${\mathbf{U}}-$space, see Theorem \[15332203\] and Corollary \[18491004\] for the ${\mathbf{LD}}({\mathcal{O}},{\mathcal{D}})$.

\[13021401\] Set $$\begin{cases}
\chi_{{\mathcal{H}}}:
End({\mathcal{H}})
\to
\prod_{x\in X}End({\mathfrak{E}}_{x}),
\\
(\forall x\in X)
(\forall w\in
End({\mathcal{H}}))
((\Pr_{x}\circ\chi_{{\mathcal{H}}})(w)
=\Pr_{x}\circ w\circ\imath_{x}),
\\
\chi\doteqdot
\chi_{\prod_{x\in X}{\mathfrak{E}}_{x}}.
\end{cases}$$ Well defined indeed by construction $\imath_{x}({\mathfrak{E}}_{x})
\subset{\mathcal{H}}$, for all $x\in X$. Finally set $$\begin{cases}
\theta:
\prod_{x\in X}End({\mathfrak{E}}_{x})
\to
End\left(\prod_{x\in X}{\mathfrak{E}}_{x}\right),
\\
(\forall x\in X)
(\forall u\in
\prod_{x\in X}End({\mathfrak{E}}_{x}))
(\Pr_{x}\circ\theta(u)=\Pr_{x}(u)\circ\Pr_{x}),
\\
\theta_{{\mathcal{H}}}:
Im(\chi_{{\mathcal{H}}})
\ni
u\mapsto
\theta(u){\upharpoonright}{\mathcal{H}}.
\end{cases}$$ Well-posed by applying [@BourA1 Prop. $4$, $n^{\circ}5$, $\S 1$,Ch. $2$].

\[10211801\] $(\forall x\in X)
(\forall u\in
\prod_{x\in X}End({\mathfrak{E}}_{x}))$ we have $(\Pr_{x}\circ\theta(u)\circ\imath_{x}
=
\Pr_{x}(u))$.

\[14152703\] The space $\prod_{x\in X}{\mathfrak{E}}_{x}$ with the product topology satisfies the request for the space ${\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}$ in Notations \[12042101\] with the choice ${\mathcal{A}}=\prod_{x\in X}{\mathcal{L}}({\mathfrak{E}}_{x})$.

$\Pr_{x}\in{\mathcal{L}}\left(
\prod_{y\in X}{\mathfrak{E}}_{y},
{\mathfrak{E}}_{x}
\right)$ by definition of the product topology, moreover $\imath_{x}\in{\mathcal{L}}\left(
{\mathfrak{E}}_{x},
\prod_{y\in X}{\mathfrak{E}}_{y}
\right)$. Indeed $\imath_{x}$ is clearly linear and by considering that for any net $\{f^{\alpha}\}_{\alpha\in D}$ and any $f$ in $\prod_{y\in X}{\mathfrak{E}}_{y}$, $\lim_{\alpha\in D}f^{\alpha}=f$ if and only if $\lim_{\alpha\in D}f^{\alpha}(y)=f(y)$ for all $y\in X$, we deduce that for any net $\{f_{x}^{\alpha}\}_{\alpha\in D}$ and any $f_{x}$ in ${\mathfrak{E}}_{x}$ such that $\lim_{\alpha\in D}f_{x}^{\alpha}=f_{x}$ we have $\lim_{\alpha\in D}
\imath_{x}(f_{x}^{\alpha})
=\imath_{x}(f_{x})$, so $\imath_{x}$ is continuous. Let $x\in X$ and $u\in\prod_{x\in X}{\mathcal{L}}({\mathfrak{E}}_{x})$ so $
\Pr_{x}(u)\circ\Pr_{x}
\in{\mathcal{L}}\left(
\prod_{y\in X}{\mathfrak{E}}_{y},
{\mathfrak{E}}_{x}
\right)$, so $(6a)$ follows by the definition of $\theta$ and [@BourGT Prp. $4$, $No 3$, $\S 2$]. Finally $(6b)$ is trivial.

The following is the main structure of the present section. For the definition and properties of locally convex final topologies see [@BourTVS $No 4$, $\S 4$].

\[10221801\] Set for all $x\in X$ $$\begin{cases}
G\doteqdot
\theta({\mathcal{A}})
{\upharpoonright}{\mathcal{H}},
\\
g_{x}:
{\mathcal{L}}({\mathfrak{E}}_{x})
\ni f_{x}\mapsto
\imath_{x}\circ f_{x}\circ\Pr_{x}
\in
End\left(
\prod_{y\in X}{\mathfrak{E}}_{y}
\right)
\\
h_{x}:
{\mathcal{L}}({\mathfrak{E}}_{x})
\ni f_{x}\mapsto
g_{x}(f_{x}){\upharpoonright}{\mathcal{H}}.
\end{cases}$$ We shall denote by **${\mathfrak{G}}$ the lcs $G$ provided with the locally convex final topology of the family of topologies $\{\tau_{x}\}_{x\in X}$ of the $\{{\mathcal{L}}({\mathfrak{E}}_{x})\}_{x\in X}$, for the family of linear mappings $\{h_{x}\}_{x\in X}$**.

\[11441302\] Set in $
\prod_{x\in X}End({\mathfrak{E}}_{x})
$ the following binary operation $\circ$. For all $x\in X$ we set $\Pr_{x}(f\circ h)\doteqdot f(x)\circ h(x)$.

It is easy to verify that ${\left\langle \prod_{x\in X}End({\mathfrak{E}}_{x}),+,\circ{\right\rangle}}$ is an algebra over ${\mathbb{K}}$ as well as ${\left\langle \prod_{x\in X}
{\mathcal{L}}({\mathfrak{E}}_{x}),+,\circ{\right\rangle}}$.

\[17141401\] $G\subset
{\mathcal{L}}\left({\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}\right)$, moreover $\theta$ is a morphism of algebras. Finally if ${\mathcal{A}}$ is a subalgebra of $\prod_{x\in X}{\mathcal{L}}({\mathfrak{E}}_{x})$ then $G$ is a subalgebra of ${\mathcal{L}}\left({\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}\right)$.

The first sentence is immediate by $(6a)$ in Notations \[12042101\]. Let $u,v\in\prod_{x\in X}{\mathcal{L}}({\mathfrak{E}}_{x})$ thus for all $x\in X$ $$\begin{aligned}
{1}
\Pr_{x}\circ\theta(u\circ v)
&=
(u(x)\circ v(x))\circ\Pr_{x}
\\
&=
u(x)\circ\Pr_{x}\circ\theta(v)
\\
&=
\Pr_{x}\circ\theta(u)\circ\theta(v),\end{aligned}$$ so $\theta(u\circ v)=\theta(u)\circ\theta(v)$, similarly we can show that $\theta$ is linear by the linearity of $\Pr_{x}$ for all $x\in X$. Thus $\theta$ is a morphism of algebras, so the last sentence of the statement follows by the first one and the fact that ${\mathcal{A}}$ is an algebra.

\[14271401\] $\theta_{{\mathcal{H}}}
\circ
\chi_{{\mathcal{H}}}
(w)
=
w\circ\imath_{x}\circ\Pr_{x}
{\upharpoonright}{\mathcal{H}}$ for all $w\in End({\mathcal{H}})$, Moreover $\theta_{{\mathcal{H}}}
(Im(\chi_{{\mathcal{H}}}))
\subset 
Dom(\chi_{{\mathcal{H}}})$ and $\chi_{{\mathcal{H}}}\circ\theta_{{\mathcal{H}}}
=Id{\upharpoonright}Im(\chi_{{\mathcal{H}}})$.

Let $w\in End({\mathcal{H}})$ thus for all $x\in X$ we have $(\Pr_{x}\circ\theta_{{\mathcal{H}}}
\circ\chi_{{\mathcal{H}}})(w)
=
\Pr_{x}(\chi_{{\mathcal{H}}}(w))\circ\Pr_{x}
{\upharpoonright}{\mathcal{H}}
=
\Pr_{x}\circ w\circ
\imath_{x}\circ\Pr_{x}{\upharpoonright}{\mathcal{H}}$ and the first sentence of the statement follows. By the first sentence and the assumption that $\imath_{x}({\mathfrak{E}}_{x})\subset{\mathcal{H}}$ we have $\theta(Im(\chi_{{\mathcal{H}}})){\upharpoonright}{\mathcal{H}}
\subset End({\mathcal{H}})$ so $\chi_{{\mathcal{H}}}\circ\theta_{{\mathcal{H}}}$ is well set. Moreover for all $x\in X$ and $u\in Im(\chi_{{\mathcal{H}}})$ we have $\Pr_{x}\left(
\chi_{{\mathcal{H}}}\left(\theta(u){\upharpoonright}{\mathcal{H}}\right)
\right)
=
\Pr_{x}\circ\theta(u)\circ\imath_{x}
=
\Pr_{x}(u)
\circ\Pr_{x}\circ\imath_{x}
=
\Pr_{x}(u)$.

\[10231801\] Let $x\in X$, then

1.  $g_{x}=\theta\circ\imath_{x}$ so $Im(h_{x})\subseteq G$;

2.  $h_{x}\in End({\mathcal{L}}({\mathfrak{E}}_{x}),G)$;

3.  $\exists\,
    h_{x}^{-1}:
    Im(h_{x})\to{\mathcal{L}}({\mathfrak{E}}_{x})$ and $$\begin{cases}
    h_{x}^{-1}=\Pr_{x}\circ\chi_{{\mathcal{H}}}
    {\upharpoonright}Im(h_{x}),
    \\
    Im(h_{x})=\{\theta(\imath_{x}(f_{x}))
    {\upharpoonright}{\mathcal{H}}
    \mid f_{x}\in
    {\mathcal{L}}({\mathfrak{E}}_{x})\}.
    \end{cases}$$

$\forall y\in X$ we have $$Pr_{y}\circ\theta(\imath_{x}(f_{x}))
=
\Pr_{y}(\imath_{x}(f_{x}))\circ\Pr_{y}
=
\begin{cases}
{\mathbf{0}}_{y},x\ne y
\\
f_{x}\circ\Pr_{x},x=y.
\end{cases}$$ Moreover $$Pr_{y}\circ g_{x}(f_{x})
=
\Pr_{y}\circ\imath_{x}
\circ 
f_{x}
\circ\Pr_{x}
=
\begin{cases}
{\mathbf{0}}_{y},x\ne y
\\
f_{x}\circ\Pr_{x},x=y.
\end{cases}$$ So the first sentence of statement $(1)$ follows. Thus $h_{x}\left({\mathcal{L}}({\mathfrak{E}}_{x})\right)
=
g_{x}\left({\mathcal{L}}({\mathfrak{E}}_{x})\right)
{\upharpoonright}{\mathcal{H}}
=
\theta\left(\imath_{x}\left(
{\mathcal{L}}({\mathfrak{E}}_{x})\right)
\right)
{\upharpoonright}{\mathcal{H}}$ so by $(6b)$ of Notations \[12042101\] the second sentence of statement $(1)$ follows. Statement $(2)$ follows by the trivial linearity of $g_{x}$ and by the second sentence of statement $(1)$.

Let $f_{x}\in{\mathcal{L}}({\mathfrak{E}}_{x})$ and $w=\imath_{x}\circ f_{x}\circ\Pr_{x}{\upharpoonright}{\mathcal{H}}$. Then by the assumption $(6)$ we have that $w\in End({\mathcal{H}})$, and $\chi_{{\mathcal{H}}}(w)=\imath_{x}(f_{x})$, indeed $\Pr_{x}(\chi_{{\mathcal{H}}}(w))
=
\Pr_{x}\circ\imath_{x}
\circ
f_{x}
\circ
\Pr_{x}\circ\imath_{x}
=
f_{x}
=
\Pr_{x}(\imath(f_{x}))$. Thus $\imath_{x}(f_{x})\in Im(\chi_{{\mathcal{H}}})$ so by Pr. \[14271401\] $\theta(\imath_{x}(f_{x})){\upharpoonright}{\mathcal{H}}
\in 
Dom(\chi_{{\mathcal{H}}})$ and $h_{x}^{-1}$ is well set. Moreover $$\begin{aligned}
{2}
(\Pr_{x}\circ\chi_{{\mathcal{H}}})\circ 
h_{x}(f_{x})
&
=
\Pr_{x}\circ\chi_{{\mathcal{H}}}
\circ\theta_{{\mathcal{H}}}(\imath_{x}(f_{x}))
\\
&=
\Pr_{x}(\imath_{x}(f_{x}))
=f_{x},\end{aligned}$$ where the first equality comes by stat. $(1)$ and by $\imath_{x}(f_{x})\in Im(\chi_{{\mathcal{H}}})$, while the second by Prop. \[14271401\]. Finally $$\begin{aligned}
{2}
g_{x}\circ\Pr_{x}\circ
\chi_{{\mathcal{H}}}(\theta(\imath_{x}(f_{x})))
&
=
g_{x}\circ\Pr_{x}(\imath_{x}(f_{x}))
\\
&=
g_{x}(f_{x})
=
\theta(\imath_{x}(f_{x})).\end{aligned}$$ Thus stat. $(3)$ follows.

\[16141302\] If ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}$ is a topological algebra for all $x\in X$ and ${\mathcal{A}}$ is an algebra then ${\mathfrak{G}}$ is a topological algebra.

Let us set for all $F\in{\mathfrak{G}}$ $L_{F}:{\mathfrak{G}}\ni H\mapsto F\circ H\in{\mathfrak{G}}$, well set ${\mathfrak{G}}$ being an algebra by Lemma \[17141401\]. Thus for all $x\in X$, $f\in{\mathcal{A}}$ and $l_{x}\in{\mathcal{L}}({\mathfrak{E}}_{x})$ $$\begin{aligned}
{2}
(L_{\theta(f)}\circ h_{x})
l_{x}
&=
L_{\theta(f)}
(\theta(\imath_{x}(l_{x}))
{\upharpoonright}{\mathcal{H}})
=
\theta(f\circ\imath_{x}(l_{x}))
{\upharpoonright}{\mathcal{H}}
\\
&=
\left[
\theta\circ\imath_{x}(f(x)\circ l_{x})
\right]
{\upharpoonright}{\mathcal{H}}
=
\left[
g_{x}(f(x)\circ l_{x})
\right]
{\upharpoonright}{\mathcal{H}}
\\
&=
h_{x}(f(x)\circ l_{x})
=
(h_{x}\circ L_{f(x)})l_{x},\end{aligned}$$ where $L_{f_{x}}:{\mathcal{L}}({\mathfrak{E}}_{x})\ni 
s_{x}\mapsto f_{x}\circ s_{x}
\in{\mathcal{L}}({\mathfrak{E}}_{x})$ for all $f_{x}\in{\mathcal{L}}({\mathfrak{E}}_{x})$. Here the first and fourth equality follow by Prop. \[10231801\], the second one by Lemma \[17141401\]. Moreover by hypothesis $L_{f(x)}$ is continuous, while $h_{x}$ is continuous by , so $L_{\theta(f)}\circ h_{x}$ is linear and continuous. Therefore $L_{\theta(f)}$ is linear and continous by . Similarly $R_{F}$ is linear and continuous, where $R_{F}:{\mathfrak{G}}\ni H\mapsto H\circ F\in{\mathfrak{G}}$, thus the statement.

\[14321401\] Set $$\begin{cases}
\Psi_{Y}^{{\mathcal{H}}}:
End({\mathcal{H}})^{Y}
\to
\prod_{x\in X}
End({\mathfrak{E}}_{x})^{Y},
\\
(\Pr_{x}\circ\Psi_{Y}^{{\mathcal{H}}})({\overline}{F})(s)
=
(\Pr_{x}\circ\chi_{{\mathcal{H}}})({\overline}{F}(s)).
\end{cases}$$ Moreover set $$\begin{cases}
\Lambda:
\prod_{x\in X}
End({\mathfrak{E}}_{x})^{Y}
\to
\left(End\left(\prod_{x\in X}{\mathfrak{E}}_{x}\right)
\right)^{Y},
\\
\Lambda(F)(s)
=
\theta(F(\cdot)(s)).
\end{cases}$$ $\forall{\overline}{F}\in End({\mathcal{H}})^{Y}$, $\forall 
F\in
\prod_{x\in X}
End({\mathfrak{E}}_{x})^{Y}
$, $\forall x\in X$ and $\forall s\in Y$, where $F(\cdot)(s)
\in\prod_{y\in X}End({\mathfrak{E}}_{x})$ such that $\Pr_{x}(F(\cdot)(s))
=
F(x)(s)$.

Finally set $$\Lambda_{{\mathcal{A}}}^{Y}
\doteqdot
\Lambda{\upharpoonright}\left\{
F\in
\prod_{x\in X}
{\mathcal{L}}({\mathfrak{E}}_{x})^{Y}
\mid
(\forall s\in Y)
(F(\cdot)(s)\in{\mathcal{A}})
\right\}.$$

\[15111401\] Let $x\in X$ and $s\in Y$, then for all ${\overline}{F}\in End({\mathcal{H}})^{Y}$

1.  $(\Pr_{x}\circ\Psi_{Y}^{{\mathcal{H}}})({\overline}{F})(s)
    =
    \Pr_{x}\circ{\overline}{F}(s)\circ\imath_{x}$;

2.  $\Psi_{Y}^{{\mathcal{H}}}
    \circ\Lambda_{{\mathcal{A}}}^{Y}=Id$;

3.  $Im(\Lambda_{{\mathcal{A}}}^{Y})
    \subset
    G^{Y}$.

Stats. $(1)$ and $(3)$ are trivial. Let $F\in Dom(\Lambda_{{\mathcal{A}}}^{Y})$ so $$\begin{aligned}
{2}
(\Pr_{x}\circ
\Psi_{Y}^{{\mathcal{H}}}
\circ
\Lambda_{{\mathcal{A}}}^{Y}
)(F)(s)
&=
(\Pr_{x}\circ\chi_{{\mathcal{H}}})(\Lambda_{{\mathcal{A}}}^{Y}(F)(s))
=
\Pr_{x}\circ\Lambda_{{\mathcal{A}}}^{Y}(F)(s)\circ\imath_{x}
\\
&=
\Pr_{x}\circ
\theta(F(\cdot)(s))
\circ\imath_{x}
=
\Pr_{x}(F(\cdot)(s))
\circ\Pr_{x}
\circ\imath_{x}
\\
&=
F(x)(s)
=
\Pr_{x}(F)(s),\end{aligned}$$ and stat. $(2)$ follows.

\[16141501\] $(\forall x\in X)
(\forall s\in Y)
(\forall{\overline}{F}\in G^{Y})$ we have $$(\Pr_{x}\circ\Psi_{Y}^{{\mathcal{H}}})
({\overline}{F})(s)
\circ\Pr_{x}
=
\Pr_{x}\circ
({\overline}{F}(s))$$

Let ${\overline}{F}\in G^{Y}$ thus $\exists\,U\in{\mathcal{A}}^{Y}$ such that ${\overline}{F}(s)=\theta(U(s))
{\upharpoonright}{\mathcal{H}}$, hence for all $x\in X,
s\in Y$ $$\begin{aligned}
{2}
(\Pr_{x}\circ\Psi_{Y}^{{\mathcal{H}}})({\overline}{F})(s)
\circ\Pr_{x}
&=
\Pr_{x}(\Psi_{Y}^{{\mathcal{H}}}({\overline}{F}))(s)
\circ
\Pr_{x}
\\
&=
\Pr_{x}\circ{\overline}{F}(s)\circ\imath_{x}
\circ
\Pr_{x},&
\text{ by Prop. \ref{15111401}}
\\
&=
(\Pr_{x}\circ
\theta(U(s)))
{\upharpoonright}{\mathcal{H}}
\circ\imath_{x}
\circ
\Pr_{x}
\\
&\doteq
(\Pr_{x}(U(s))
\circ\Pr_{x})
{\upharpoonright}{\mathcal{H}}
\circ\imath_{x}
\circ
\Pr_{x}
\\
&=
\Pr_{x}(U(s))
\circ\Pr_{x}
{\upharpoonright}{\mathcal{H}}
\\
&\doteq
\Pr_{x}
\circ
\theta(U(s))
{\upharpoonright}{\mathcal{H}}
\\
&=
\Pr_{x}
\circ
({\overline}{F}(s)).\end{aligned}$$

\[11221801\] Let $x\in X$ $$\begin{cases}
I_{x}:Hom({\mathcal{L}}({\mathfrak{E}}_{x}),{\mathbb{K}})
\to
Hom\left(
\prod_{y\in X}{\mathcal{L}}({\mathfrak{E}}_{y}),{\mathbb{K}}\right),
\\
I_{x}(t_{x})\doteqdot t_{x}\circ\Pr_{x}.
\end{cases}$$

\[12001801\] Let $x\in X$ thus

1.  $(\forall t_{x}\in Hom({\mathcal{L}}({\mathfrak{E}}_{x}),{\mathbb{K}}))
    (\forall y\in X)$ we have $$\begin{cases}
    I_{x}(t_{x})\circ\chi_{{\mathcal{H}}}\circ h_{y}
    =
    t_{x},
    x=y
    \\
    I_{x}(t_{x})\circ\chi_{{\mathcal{H}}}\circ h_{y}
    ={\mathbf{0}},
    x\neq y;
    \end{cases}$$

2.  $(\forall t_{x}\in
    {\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}^{*})
    (I_{x}(t_{x})\circ\chi_{{\mathcal{H}}}{\upharpoonright}{\mathfrak{G}}
    \in{\mathfrak{G}}^{*})$

Let $x\in X$ and $t_{x}\in
Hom({\mathcal{L}}({\mathfrak{E}}_{x}),{\mathbb{K}})$ thus for all $y\in X$ and $f_{y}\in{\mathcal{L}}({\mathfrak{E}}_{y})$ we have $$\begin{aligned}
{1}
I_{x}(t_{x})\circ\chi_{{\mathcal{H}}}\circ h_{y}
(f_{y})
&=
t_{x}\circ\Pr_{x}\circ\chi_{{\mathcal{H}}}
(\imath_{y}\circ f_{y}\circ\Pr_{y}{\upharpoonright}{\mathcal{H}})
\\
&=
t_{x}\circ
(\Pr_{x}\circ\imath_{y}\circ f_{y}
\circ\Pr_{y}\circ\imath_{x}),\end{aligned}$$ and stat. $(1)$ follows. Stat. $(2)$ follows by stat. $(1)$ and .

The following is the first main result of this section.

\[15251401\] We have

1.  $\Psi_{Y}^{{\mathcal{H}}}
    \in
    Hom({\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu),
    \prod_{x\in X}
    {\mathfrak{L}}_{1}(Y,
    {\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}},\mu))$;

2.  $(\forall x\in X)(\forall s\in Y)
    (\forall{\overline}{F}\in{\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu))
    $ $$\int\Pr_{x}(\Psi_{Y}^{{\mathcal{H}}}({\overline}{F}))(s)
    \,d\mu(s)
    =
    \Pr_{x}
    \circ
    \left[
    \int{\overline}{F}(s)\,d\mu(s)
    \right]
    \circ\imath_{x}.$$

Let $x\in X$, set $$\Delta_{x}:G\ni f
\mapsto
\Pr_{x}\circ f\circ\imath_{x}
\in
{\mathcal{L}}({\mathfrak{E}}_{x}).$$ $\Delta_{x}$ is well-defined by Lemma \[17141401\]. By applying $\Delta_{x}\in{\mathcal{L}}
({\mathfrak{G}},
{\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}})$ if and only if $(\forall y\in X)
(\Delta_{x}\circ h_{y}\in
{\mathcal{L}}({\left\langle {\mathcal{L}}({\mathfrak{E}}_{y}),\tau_{y}{\right\rangle}},
{\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}))$. Moreover $\forall y\in X$ and $\forall f_{y}\in{\mathcal{L}}({\mathfrak{E}}_{y})$ we have $$(\Delta_{x}\circ h_{y})(f_{y})
=
\Pr_{x}\circ\imath_{y}\circ f_{y}
\circ\Pr_{y}\circ\imath_{x},$$ so $$\begin{cases}
\Delta_{x}\circ h_{y}=Id, x=y
\\
\Delta_{x}\circ h_{y}={\mathbf{0}},
x\neq y.
\end{cases}$$ In any case $\Delta_{x}\circ h_{y}\in
{\mathcal{L}}({\left\langle {\mathcal{L}}({\mathfrak{E}}_{y}),\tau_{y}{\right\rangle}},
{\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}})$, thus $$\Delta_{x}\in{\mathcal{L}}
({\mathfrak{G}},
{\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}})$$ hence $$\label{13351801}
(\forall t_{x}\in
{\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}^{*})
(t_{x}\circ
\Delta_{x}
\in{\mathfrak{G}}^{*}).$$ Therefore $$\begin{aligned}
{1}
t_{x}\left(
\Pr_{x}\circ\left[
\int{\overline}{F}(s)\,d\mu(s)
\right]
\circ\imath_{x}
\right)
&=
(t_{x}\circ\Delta)
\left(\int{\overline}{F}(s)\,d\mu(s)\right)
\\
&=
\int(t_{x}\circ\Delta)({\overline}{F}(s))\,d\mu(s)
\\
&=
\int t_{x}\left(\Pr_{x}\circ
{\overline}{F}(s)\circ\imath_{x}\right)\,d\mu(s)
\\
&=
\int
t_{x}\left(
(\Pr_{x}\circ\Psi_{Y}^{{\mathcal{H}}})({\overline}{F})(s)
\right)
\,d\mu(s),\end{aligned}$$ where the second equality comes by and , while the last one comes by Proposition \[15111401\].

\[11121501\] Let $Z$ be a topological vector space set $$\begin{cases}
{\mathbf{ev}}_{Z}
\in 
Hom(Z,Hom({\mathcal{L}}(Z),Z),
\\
(\forall v\in Z)
(\forall f\in{\mathcal{L}}(Z))
({\mathbf{ev}}_{Z}(v)(f))
\doteqdot
f(v)).
\end{cases}$$ Moreover set $\eta
\doteqdot
{\mathbf{ev}}_{{\mathcal{H}}}$ and $\forall x\in X$ set ${\varepsilon}_{x}
\doteqdot
{\mathbf{ev}}_{{\mathfrak{E}}_{x}}$.

\[11011501\] Let $D\subseteq{\mathcal{H}}$ thus $(A)\Rightarrow(B)$, where

(A)

:   $(\forall x\in X)(\forall v_{x}\in\Pr_{x}(D))
    ({\varepsilon}_{x}(v_{x})\in{\mathcal{L}}
    ({\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}},{\mathfrak{E}}_{x}))$;

(B)

:   $(\forall v\in D)
    (\eta(v)\in{\mathcal{L}}
    ({\mathfrak{G}},{\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}))$.

Let $y\in X$ thus for all $v\in{\mathcal{H}}$ $$\eta(v)\circ h_{y}
=
\imath_{y}\circ{\varepsilon}_{y}(\Pr_{y}(v)).$$ Hence by $(A)$ and the fact that by construction $\imath_{y}$ is continuous with respect to the topology ${\mathfrak{T}}$ we have for all $v\in D$ $$\eta(v)\circ g_{y}
\in
{\mathcal{L}}\left(
{\left\langle {\mathcal{L}}({\mathfrak{E}}_{y}),\tau_{y}{\right\rangle}},
{\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}
\right).$$ Thus $(B)$ follows by the universal property of any locally final topology, cf. [@BourTVS $(ii)$ of Prop. $5$, $N\,4$, $\S 4$ Ch $2$].

The following is the second main result of the section

\[16291501\] Let $D\subseteq{\mathcal{H}}$ and assume $(A)$ of Lemma \[11011501\]. Then $(\forall{\overline}{F}
\in{\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu))
(\forall x\in X)
(\forall v\in D)$ $$\label{15281901}
\int
{\left\langle \Pr_{x}(\Psi_{Y}^{{\mathcal{H}}}
({\overline}{F}))(s),v(x){\right\rangle}}_{x}
\,d\mu(s)
=
{\left\langle \int{\overline}{F}(s)\,d\mu(s),v{\right\rangle}}(x).$$ Here the integral in the left-side is with respect to the $\mu$ and the lct on ${\mathfrak{E}}_{x}$, while the integral in the right-side is with respect to the $\mu$ and the lct on ${\mathfrak{G}}$.

$(\forall{\overline}{F}
\in{\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu))
(\forall x\in X)
(\forall v\in D)$ we have $$\begin{aligned}
{1}
\Pr_{x}\circ
\left[
\int{\overline}{F}(s)\,d\mu(s)
\right](v)
&=
(\Pr_{x}\circ\eta(v))
\left(\int{\overline}{F}(s)\,d\mu(s)\right)
\\
&=
\int
(\Pr_{x}\circ\eta(v))
({\overline}{F}(s))
\,d\mu(s)
\\
&=
\int
(\Pr_{x}\circ{\overline}{F}(s))(v)\,
d\mu(s)
\\
&=
\int
\Pr_{x}(\Psi_{Y}^{{\mathcal{H}}}({\overline}{F}))(s)
(v(x))
\,d\mu(s).\end{aligned}$$ Here in the second equality we applied and the fact that $\Pr_{x}\circ\eta(v)
\in{\mathcal{L}}({\mathfrak{G}},{\mathfrak{E}}_{x})$ because of Lemma \[11011501\] and the linearity and continuity of $\Pr_{x}$ with respect to the topology ${\mathfrak{T}}$. Finally in the last equality we used Prop. \[16141501\].

The following is the main result of this section

\[15332203\] Let $D\subseteq{\mathcal{H}}$ and assume $(A)$ of Lemma \[11011501\]. Then $(\forall{\overline}{F}
\in{\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu))
(\forall x\in X)
(\forall v\in D)$ $$\label{14342203}
\boxed{
{\left\langle \int
\Pr_{x}(\Psi_{Y}^{{\mathcal{H}}}({\overline}{F}))(s)
\,d\mu(s),v(x){\right\rangle}}_{x}
=
{\left\langle \int{\overline}{F}(s)\,d\mu(s),v{\right\rangle}}(x).
}$$ Equivalently ${\mathfrak{G}}$ is a ${\mathbf{U}}-$space with respect to $\{{\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}\}_{x\in X}$, ${\mathfrak{T}}$ and $D$. Here the integral in the left-side is with respect to the $\mu$ and the lct on ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}$.

By $(A)$ of Lemma \[11011501\], stat.$(1)$ of Th. \[15251401\] and we have $(\forall{\overline}{F}
\in{\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu))
(\forall x\in X)
(\forall v\in D)$ $$\int
{\left\langle \Pr_{x}
(\Psi_{Y}^{{\mathcal{H}}}({\overline}{F}))(s),v(x){\right\rangle}}_{x}
\,d\mu(s)
=
{\left\langle \int
\Pr_{x}(\Psi_{Y}^{{\mathcal{H}}}({\overline}{F}))(s)
\,d\mu(s),v(x){\right\rangle}}_{x},$$ hence the statement follows by Theorem \[16291501\].

\[16082203\] By and stat.$(2)$ of Th. \[15251401\] $(\forall{\overline}{F}
\in{\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu))
(\forall x\in X)
(\forall v\in D)
$ $${\left\langle \int{\overline}{F}(s)\,d\mu(s),v{\right\rangle}}(x)
=
{\left\langle \int{\overline}{F}(s)\,d\mu(s),\imath_{x}(v(x)){\right\rangle}}(x).$$ Thus for all $v,w\in D$ and $x\in X$ $$v(x)=w(x)
\Rightarrow
{\left\langle \int{\overline}{F}(s)\,d\mu(s),v{\right\rangle}}(x)
=
{\left\langle \int{\overline}{F}(s)\,d\mu(s),w{\right\rangle}}(x).$$

\[15111901\] Let ${\mathcal{S}}\in\prod_{x\in X}
2^{Bounded({\mathfrak{E}}_{x})}$ and ${\mathcal{D}}$ such that $$\label{15391901a}
\begin{cases}
N(x)
\doteqdot
\bigcup_{l_{x}\in L_{x}}
B_{l_{x}}^{x}
\text{ is total in ${\mathfrak{E}}_{x}$},
\forall x\in X,
\\
{\mathcal{D}}\subseteq
{\mathcal{H}}
\cap
\prod_{x\in X}N(x),
\end{cases}$$ where ${\mathcal{S}}(x)
=\{B_{l_{x}}^{x}\mid l_{x}\in L_{x}\}$. Assume that for all $x\in X$ the topology $\tau_{x}$ is generated by the set of seminorms $\{
p_{(l_{x},j_{x})}^{x}
\mid
(l_{x},j_{x})\in
L_{x}\times J_{x}\}$, where [^16] $$\label{15391901b}
p_{(l_{x},j_{x})}^{x}:
{\mathcal{L}}({\mathfrak{E}}_{x})\ni
f_{x}
\mapsto
\sup_{w\in B_{l_{x}}^{x}}
\nu_{j_{x}}^{x}(f_{x}w)
\in{\mathbb {R}}^{+}.$$ Then

1.  $(A)$ of Lemma \[11011501\] for $D={\mathcal{D}}$;

2.  holds and ${\mathfrak{G}}$ is a ${\mathbf{U}}-$space with respect to $\{{\mathcal{L}}_{{\mathcal{S}}(x)}({\mathfrak{E}}_{x})\}_{x\in X}$, ${\mathfrak{T}}$ and ${\mathcal{D}}$.

By request we have that the lcs ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}$ is Hausdorff so the position is well-set. By construction $(\forall x\in X)
(\forall v_{x}\in D(x))
(\exists\,{\overline}{l}_{x}\in L_{x})
(v_{x}\in B_{{\overline}{l}_{x}}^{x})$, so $(\forall f_{x}\in{\mathcal{L}}({\mathfrak{E}}_{x}))
(\forall j_{x}\in J_{x})$ $$\begin{aligned}
{1}
\nu_{j_{x}}^{x}
({\varepsilon}_{x}(v_{x})f_{x})
&=
\nu_{j_{x}}^{x}(f_{x}(v_{x}))
\\
&\leq
p_{({\overline}{l}_{x},j_{x})}^{x}(f_{x}),\end{aligned}$$ hence statement $(1)$ by . Statement $(2)$ follows by statement $(1)$, Theorem \[16291501\] and Theorem \[15332203\] respectively.

\[${\mathbf{LD}}({\mathcal{O}},{\mathcal{D}})$\] \[18491004\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure satisfying and $
\Gamma(\pi)
\cap{\mathcal{H}}
\cap
\prod_{x\in X}{\mathcal{B}}_{B}^{x}
\ne\emptyset
$. Set $$\label{21241004}
\begin{cases}
{\mathcal{O}}\subseteq\Gamma(\rho)
\\
{\mathcal{D}}\subseteq
\Gamma(\pi)\cap{\mathcal{H}}
\cap
\prod_{x\in X}{\mathcal{B}}_{B}^{x}
\end{cases}$$ If $\exists\,
{\mathcal{F}}\subset
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi_{{\mathbb {R}}^{+}}^{{\mathcal{H}}}({\mathcal{F}})={\mathcal{O}}$ then holds.

In particular if $\exists\,{\mathcal{F}}\subset
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi_{{\mathbb {R}}^{+}}^{{\mathcal{H}}}({\mathcal{F}})={\mathcal{O}}$ then $${\left\langle {\mathfrak{G}},{\mathcal{D}}{\right\rangle}}
\subseteq
\Gamma(\pi)
\Rightarrow
{\mathbf{LD}}({\mathcal{O}},{\mathcal{D}}).$$ Here ${\mathcal{B}}_{B}^{x}$, for all $x\in X$, is defined in .

By statement $(2)$ of Cor. \[15111901\], Pr. \[14492503\] and Rm. \[13390104\].

\[21211004\] Note that if ${\mathcal{E}}\subset\Theta$, as for example for the positions taken in Rm. \[21500412b\], we have ${\mathcal{E}}\subset\prod_{x\in X}{\mathcal{B}}_{B}^{x}$. Hence if ${\mathcal{E}}\subseteq{\mathcal{H}}$ we have ${\mathcal{E}}
\subseteq
\Gamma(\pi)
\cap
{\mathcal{H}}
\cap
\prod_{x\in X}{\mathcal{B}}_{B}^{x}$.

By the previous remark, Cor. \[18491004\] and Thm. \[17301812b\] we can state

\[12080805\] Let us assume the hypotheses of Thm. \[17301812b\] made exception for the $(i)$ replaced by the following one: ${\mathcal{E}}\subseteq{\mathcal{H}}$ and $\exists\,
{\mathcal{F}}\subset
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi_{{\mathbb {R}}^{+}}^{{\mathcal{H}}}({\mathcal{F}})=\Gamma(\rho)$ and $${\left\langle {\mathfrak{G}},{\mathcal{E}}{\right\rangle}}
\subseteq
\Gamma(\pi).$$ Then all the statements of Thm. \[17301812b\] hold

###  Uniform Convergence over ${\mathcal{K}}\in Compact({\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}})$. 

In this subsection we assume given the following data

1.  a ${\mathfrak{V}}$ Banach bundle, a $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{M}},X,Y{\right\rangle}}$ where $\Theta$ is defined in , where we denote ${\mathfrak{W}}
    \doteqdot
    {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ and ${\mathfrak{V}}
    \doteqdot
    {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\{\|\cdot\|\}{\right\rangle}}$;

2.  a Banach space ${\left\langle {\mathcal{H}},\|\cdot\|_{{\mathcal{H}}}{\right\rangle}}$ such that ${\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}$ satisfies $(6)$ of Notations \[12042101\], where ${\mathfrak{T}}$ is the topology induced by the norm $\|\cdot\|_{{\mathcal{H}}}$ and $\tau_{x}$ is such that ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}
    =
    {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$ for every $x\in X$;

3.  ${\mathcal{A}}$ as in $(6)$ of Notations \[12042101\];

4.  ${\mathfrak{G}}$, $\Psi_{Y}^{{\mathcal{H}}}$ and $\Lambda_{{\mathcal{A}}}^{Y}$ as defined in Def. \[10221801\] and Def. \[14321401\] respectively.

The proof of the following Lemma is an adaptation to the present framework of the proof of [@cho Prop. $5.13$].

\[19260304\] Let ${\mathcal{U}}\in\prod_{x\in X}{\mathfrak{M}}_{x}$ and $x_{\infty}\in X$ moreover assume that

1.  ${\mathcal{E}}\subseteq{\mathcal{H}}
    \subseteq\prod_{x\in X}^{b}{\mathfrak{E}}_{x}$ such that $(\exists\,a>0)(\forall f\in{\mathcal{H}})
    (\|f\|_{\sup}\leq a\|f\|_{{\mathcal{H}}})$, where $\|f\|_{\sup}\doteqdot
    \sup_{x\in X}\|f(x)\|_{x}$;

2.  $\exists\,F\in\Gamma(\rho)$ such that $F(x_{\infty})={\mathcal{U}}(x_{\infty})$ and $\{F(\cdot)(s)\mid s\in Y\}
    \subseteq{\mathcal{A}}$

3.  $\{{\mathcal{U}}(\cdot)(s)\mid s\in Y\}
    \subseteq{\mathcal{A}}$;

4.  $\{{\overline}{F}(s)\mid s\in Y\}$ and $\{{\overline}{{\mathcal{U}}}(s)\mid s\in Y\}$ are equicontinuous as subsets of ${\mathcal{L}}({\left\langle {\mathcal{H}},\|\cdot\|_{{\mathcal{H}}}{\right\rangle}})$, where ${\overline}{{\mathcal{U}}}\doteqdot\Lambda_{{\mathcal{A}}}^{Y}({\mathcal{U}})$. and ${\overline}{F}\doteqdot\Lambda_{{\mathcal{A}}}^{Y}(F)$.

Then $(A)\Leftrightarrow(B)$ where

:   ${\mathcal{U}}\in\Gamma^{x_{\infty}}(\rho)$;

:   For all ${\mathcal{K}}\in Compact({\mathcal{H}})$ such that ${\mathcal{K}}\subseteq{\mathcal{E}}$ and for all $K\in Compact(Y)$ $$\lim_{z\to x_{\infty}}
    \sup_{s\in K}
    \sup_{v\in{\mathcal{K}}}
    \left\|
    {\mathcal{U}}(z)(s)v(z)
    -
    F(z)(s)v(z)
    \right\|
    =0.$$

We shall prove only $(A)\Rightarrow(B)$, indeed the other implication follows by $(3)\Rightarrow(4)$ of Lemma \[15482712\]. So assume $(A)$ to be true. In this proof let us set $B({\mathcal{H}})\doteqdot
{\mathcal{L}}({\left\langle {\mathcal{H}},\|\cdot\|_{{\mathcal{H}}}{\right\rangle}})$, moreover $\Psi\doteqdot\Psi_{Y}^{{\mathcal{H}}}$ and $\Lambda\doteqdot\Lambda_{{\mathcal{A}}}^{Y}$, moreover set ${\overline}{F}\doteqdot\Lambda_{{\mathcal{A}}}^{Y}(F)$ for every $F\in\Gamma(\rho)$; thus by stat. $(2)$ of Pr. \[15111401\] $\Psi({\overline}{F})=F$ and $\Psi({\overline}{{\mathcal{U}}})={\mathcal{U}}$. Hence by Pr. \[16141501\] for all $v\in{\mathcal{E}}$ $F\in\Gamma(\rho)$, $z\in X$ and $s\in Y$ $$\label{20160304}
{\mathcal{U}}(z)(s)v(z)
=
({\overline}{{\mathcal{U}}}v)(z),\,
F(z)(s)v(z)
=
({\overline}{F}v)(z).$$ By $(A)$ and implication $(4)\Rightarrow(3)$ of Lemma \[15482712\] we have for all $K\in Compact(Y)$ and $v\in{\mathcal{E}}$ $$\label{20150304}
\lim_{z\to x_{\infty}}
\sup_{s\in K}
\left\|
{\mathcal{U}}(z)(s)v(z)
-
F(z)(s)v(z)
\right\|
=0.$$ Fix ${\mathcal{K}}\in Compact({\mathcal{H}})$ such that ${\mathcal{K}}\subseteq{\mathcal{E}}$, $f\in{\mathcal{K}}$ and ${\varepsilon}>0$, thus by and there exists $U$ neighbourhood of $x_{\infty}$ such that $$\label{20170304}
\sup_{s\in K}
\sup_{z\in U}
\left\|
\left[({\overline}{{\mathcal{U}}}(s)-{\overline}{F}(s))f\right](z)
\right\|
\leq{\varepsilon}/2.$$ Define $$\begin{cases}
M
\doteqdot
\max\{
\sup_{s\in Y}
\|{\overline}{F}(s)\|_{B({\mathcal{H}})},
\,
\sup_{s\in Y}
\|{\overline}{{\mathcal{U}}}(s)\|_{B({\mathcal{H}})}
\}
\\
\eta\doteqdot{\varepsilon}/4aM
\\
{\mathfrak{U}}(f)
\doteqdot
\{g\in{\mathcal{K}}\mid\|f-g\|_{{\mathcal{H}}}<\eta\}.
\end{cases}$$ Thus for all $g\in{\mathfrak{U}}(f)$ $$\begin{aligned}
{1}
\sup_{z\in U}
\sup_{s\in K}
\left\|
{\mathcal{U}}(z)(s)g(z)
-
F(z)(s)g(z)
\right\|
&
=
\\
\sup_{s\in K}
\sup_{z\in U}
\left\|
\left[({\overline}{{\mathcal{U}}}(s)-{\overline}{F}(s))g\right](z)
\right\|
&
\leq
\\
\sup_{s\in K}
\sup_{z\in U}
\left\|
\left[({\overline}{{\mathcal{U}}}(s)-{\overline}{F}(s))f\right](z)
\right\|
+
\sup_{s\in K}
\sup_{z\in U}
\left\|
{\overline}{{\mathcal{U}}}(s)(g-f)(z)
\right\|
+
\sup_{s\in K}
\sup_{z\in U}
\left\|
F(s)(g-f)(z)
\right\|
&
\leq
\\
{\varepsilon}/2
+
a
\sup_{s\in K}
\left\|
{\overline}{{\mathcal{U}}}(s)(g-f)
\right\|_{{\mathcal{H}}}
+
a
\sup_{s\in K}
\left\|
F(s)(g-f)
\right\|_{{\mathcal{H}}}
&
\leq
\\
{\varepsilon}/2
+
2aM
\left\|g-f\right\|_{{\mathcal{H}}}
&
<
{\varepsilon}.\end{aligned}$$ Therefore $(B)$ follows by considering that $\{{\mathfrak{U}}(f)\mid f\in{\mathcal{K}}\}$ is an open cover of the compact ${\mathcal{K}}$. Indeed let for example $\{{\mathfrak{U}}(f_{i})\mid i=1,...,n\}$ a finite subcover of ${\mathcal{K}}$ thus by setting $W\doteqdot\bigcap_{i=1}^{n}U_{n}$ with obvious meaning of $U_{i}$, we have $$\sup_{z\in W}
\sup_{s\in K}
\sup_{g\in{\mathcal{K}}}
\left\|
{\mathcal{U}}(z)(s)g(z)
-
F(z)(s)g(z)
\right\|
<
{\varepsilon}.$$

We can set ${\mathcal{H}}=\prod_{x\in X}^{b}{\mathfrak{E}}_{x}$ with the usual norm $\|\cdot\|_{\sup}$.

\[${\mathcal{K}}-$Uniform Convergence\] \[10581004\] Let ${\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\|\cdot\|{\right\rangle}}$ be a Banach bundle. Let $x_{\infty}\in X$ and $
{\mathcal{U}}_{0}
\in\prod_{x\in X_{0}}
{\mathcal{C}_{}
\left({\mathbb {R}}^{+},B_{s}({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x}$ for all $x\in X_{0}$. Assume that

1.  $D(T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$;

2.  ${\mathfrak{V}}$ and ${\mathfrak{W}}$ satisfy ;

3.  $\exists\lambda_{0}>0$ (respectively $\exists\lambda_{0}>0,
    \lambda_{1}<0$) such that the range ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$, (respectively the ranges ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ and ${\mathcal{R}}(\lambda_{1}-T_{x_{\infty}})$ are dense in ${\mathfrak{E}}_{x_{\infty}}$);

4.  $
    {\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{z})}}
    ({\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z}))
    \subseteq
    {\mathfrak{M}}_{z}$ (respectively ${\mathbf{U}}_{is}
    ({\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z}))
    \subseteq
    {\mathfrak{M}}_{z}$) for all $z\in X$;

5.  ${\mathcal{E}}\subseteq{\mathcal{H}}
    \subseteq\prod_{x\in X}^{b}{\mathfrak{E}}_{x}$

6.  $X$ is metrizable;

7.  $\exists\,{\mathcal{F}}\subset
    \bigcap_{\lambda>0}
    {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi_{{\mathbb {R}}^{+}}^{{\mathcal{H}}}({\mathcal{F}})=\Gamma(\rho)$;

8.  $(\exists\,F\in\Gamma(\rho))
    (F(x_{\infty})={\mathcal{U}}(x_{\infty}))$ such that

    1.  ${\left\langle \int{\overline}{F}(s)\,d\mu_{\lambda}(s),{\mathcal{E}}{\right\rangle}}\subseteq\Gamma(\pi)$, for all $\lambda>0$;

    2.  $(\forall v\in{\mathcal{E}})
        (\exists\,\phi\in\Phi)$ s.t. $\phi_{1}(x_{\infty})
        =
        v(x_{\infty})$ and $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X
        \mid
        \lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ we have that $\{
        {\mathcal{U}}(z_{n})(\cdot)\phi_{1}(z_{n})
        -
        F(z_{n})(\cdot)v(z_{n})
        \}_{n\in{\mathbb{N}}}$ is a bounded equicontinuous sequence.

Then $$\label{19351004}
{\mathcal{U}}
\in\Gamma^{x_{\infty}}(\rho).$$ Furthermore if

1.  $(\exists\,a>0)(\forall f\in{\mathcal{H}})
    (\|f\|_{\sup}\leq a\|f\|_{{\mathcal{H}}})$,

2.  $\{F(\cdot)(s)\mid s\in{\mathbb {R}}^{+}\}
    \subseteq{\mathcal{A}}$ and $\{{\mathcal{U}}(\cdot)(s)\mid s\in{\mathbb {R}}^{+}\}
    \subseteq{\mathcal{A}}$;

3.  $\{{\overline}{F}(s)\mid s\in{\mathbb {R}}^{+}\}$ and $\{{\overline}{{\mathcal{U}}}(s)\mid s\in{\mathbb {R}}^{+}\}$ are equicontinuous as subsets of ${\mathcal{L}}({\left\langle {\mathcal{H}},\|\cdot\|_{{\mathcal{H}}}{\right\rangle}})$.

Then for all ${\mathcal{K}}\in Compact({\mathcal{H}})$ such that ${\mathcal{K}}\subseteq{\mathcal{E}}$ and for all $K\in Compact({\mathbb {R}}^{+})$ $$\label{19371004}
\lim_{z\to x_{\infty}}
\sup_{s\in K}
\sup_{v\in{\mathcal{K}}}
\left\|
{\mathcal{U}}(z)(s)v(z)
-
F(z)(s)v(z)
\right\|
=0.$$ Here $D(T_{x_{\infty}})$ is defined as in Notations \[15411512b\] with ${\mathcal{T}}_{0}$ and $\Phi$ given in . While ${\mathcal{U}}
\in\prod_{x\in X}
{\mathfrak{M}}_{x}$ such that ${\mathcal{U}}{\upharpoonright}X_{0}\doteqdot
{\mathcal{U}}_{0}$ and ${\mathcal{U}}(x_{\infty})$ is the semigroup on ${\mathfrak{E}}_{x_{\infty}}$ generated by $T_{x_{\infty}}$ operator defined in . Moreover $\|f\|_{\sup}\doteqdot
\sup_{x\in X}\|f(x)\|_{x}$, while ${\overline}{{\mathcal{U}}}\doteqdot\Lambda_{{\mathcal{A}}}^{Y}({\mathcal{U}})$ and ${\overline}{F}\doteqdot\Lambda_{{\mathcal{A}}}^{Y}(F)$.

By hyp. $(7)$ and st. $(1)$ of Th. \[15251401\], follows. Moreover follows by hyp. $(5)$, and Rm. \[21211004\]. Hence by hyps. $(7-8a)$, and Cor. \[18491004\] follows the ${\mathbf{LD}}(\{F\},{\mathcal{E}})$. Then follows by Th. \[17301812b\]. follows by and Lm. \[19260304\].

\[23461004\] By st.$(2)$ of Pr. \[15111401\] hyp. $(7)$ is equivalent to the following one $\Lambda_{{\mathcal{A}}}^{{\mathbb {R}}^{+}}(\Gamma(\rho))
\subseteq
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$. In any case the form in hyp. $(7)$ has the advantage to be considered as a tool for constructing $\Gamma(\rho)$. Finally note that $${\left\langle {\mathfrak{G}},{\mathcal{E}}{\right\rangle}}\subseteq\Gamma(\pi)
\Rightarrow
(8a).$$

###  ${\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}$ as Direct Integral of a Continuous Field of left-Hilbert and associated left-von Neumann Algebras.

#### Ex1

Assume that ${\mathfrak{V}}
=
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ is a continuous field of left-Hilbert algebras on $X$. Let ${\mathcal{H}}$ be the direct integral of ${\mathfrak{V}}$ with respect to some finite Radon measure on $X$ and ${\mathcal{B}}\subset{\mathcal{H}}$ a linear space, set $${\mathcal{A}}({\mathcal{B}})
\doteqdot
\left\{
X\ni x\mapsto L_{a(x)}
\mid
a\in{\mathcal{B}}\right\},$$ where $L_{a_{x}}\in B({\mathfrak{E}}_{x})$ for any $a_{x}\in{\mathfrak{E}}_{x}$, is the left multiplication on the left-Hilbert algebra ${\mathfrak{E}}_{x}$. Then ${\mathcal{H}}$ and ${\mathcal{A}}({\mathcal{B}})$ satisfies the requirements in Notations \[12042101\], moreover $$\label{14502904}
\boxed{
G({\mathcal{B}})
\doteqdot
\theta({\mathcal{A}}({\mathcal{B}})){\upharpoonright}{\mathcal{H}}
=
L_{{\mathcal{B}}},}$$ where $L_{a}\in B({\mathcal{H}})$ for any $a\in{\mathcal{H}}$, is the left multiplication on the left-Hilbert algebra ${\mathcal{H}}$. If every ${\mathfrak{E}}_{x}$ is unital then ${\mathcal{H}}$ is unital, thus $L_{(\cdot)}$ is an injective (isometric) map of ${\mathcal{H}}$ into $B({\mathcal{H}})$. Therefore under this additional requirement we can take the following identification $$G({\mathcal{B}})\simeq{\mathcal{B}}\text{ as linear spaces}.$$ Let ${\mathbf{H}}\doteqdot
\{{\mathbf{H}}^{i}\in\prod_{x\in X}{\mathfrak{E}}_{x}
\}_{i=0}^{2}$ such that ${\mathbf{H}}_{x}^{0}$ is a left Hilbert subalgebra of ${\mathfrak{E}}_{x}$, while ${\mathbf{H}}_{x}^{k}$ is a linear subspace of ${\mathbf{H}}_{x}^{0}$, for all $k=1,2$ and $x\in X$. Set $$\label{14582904}
\begin{cases}
\Gamma(\pi,{\mathbf{H}})
\doteqdot
\left\{
\sigma\in{\mathcal{H}}
\mid
(\forall x\in X)
(\sigma(x)\in{\mathbf{H}}_{x}^{0})
\right\}
\\
{\mathcal{D}}_{{\mathbf{H}}}
\doteqdot
\left\{
\sigma\in{\mathcal{H}}
\mid
(\forall x\in X)
(\sigma(x)\in{\mathbf{H}}_{x}^{1})
\right\}
\\
{\mathcal{B}}_{{\mathbf{H}}}
\doteqdot
\left\{
\sigma\in{\mathcal{H}}
\mid
(\forall x\in X)
(\sigma(x)\in{\mathbf{H}}_{x}^{2})
\right\}.
\end{cases}$$ Thus $\Gamma(\pi,{\mathbf{H}})$ is a left Hilbert subalgebra of ${\mathcal{H}}$ and ${\mathcal{B}}_{{\mathbf{H}}}, {\mathcal{D}}_{{\mathbf{H}}}$ are linear subspaces of $\Gamma(\pi,{\mathbf{H}})$, so $$\label{14512904}
L_{{\mathcal{B}}_{{\mathbf{H}}}}({\mathcal{D}}_{{\mathbf{H}}})
\subseteq
\Gamma(\pi,{\mathbf{H}}).$$ By and follows that for all $\sigma\in{\mathcal{B}}_{{\mathbf{H}}}$, $\eta\in{\mathcal{D}}_{{\mathbf{H}}}$ and $y\in X$ $$\label{15172904}
\boxed{
\begin{cases}
{\left\langle G({\mathcal{B}}_{{\mathbf{H}}}),{\mathcal{D}}_{{\mathbf{H}}}{\right\rangle}}
\subseteq
\Gamma(\pi,{\mathbf{H}}),
\\
{\left\langle \theta\left(
x\mapsto L_{\sigma(x)}\right),\eta{\right\rangle}}(y)
=
\sigma(y)\eta(y).
\end{cases}
}$$

#### Ex2

Let us consider now the continuous field of left-von Neumann algebras associated to the fixed field of Hilbert algebras, and by abusing of language, let us denote it with the same symbol ${\mathfrak{V}}
=
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$, as well as ${\mathcal{H}}$ will denote the associated direct integral with respect to some finite Radon measure on $X$. Let $\Delta_{x}$ be the modular operator associated to the Hilbert algebra ${\mathfrak{E}}_{x}$ and $\sigma_{x}$ the corresponding modular group. Thus we can set $$\begin{cases}
{\mathcal{A}}_{\Delta}
\doteqdot\{
S_{t}:
X\ni x\mapsto
\sigma_{x}(t)\in 
Aut({\mathfrak{E}}_{x})
\mid t\in{\mathbb {R}}\}
\\
G_{\Delta}\doteqdot
\theta({\mathcal{A}}_{\Delta}){\upharpoonright}{\mathcal{H}}
\\
\Sigma_{t}
\doteqdot
\theta(S_{t}){\upharpoonright}{\mathcal{H}},\,
t\in{\mathbb {R}}.
\end{cases}$$ Note that for every $t\in{\mathbb {R}}$, $v\in{\mathcal{H}}$ and $x\in X$ $$\Sigma_{t}(v)(x)
=\sigma_{x}(t)(v(x)).$$ Now if we set $$\Gamma(\pi)
\doteqdot{\mathcal{H}}$$ for any linear subspace ${\mathcal{D}}$ of ${\mathcal{H}}$ we have $$\boxed{
{\left\langle G_{\Delta},{\mathcal{D}}{\right\rangle}}
\subseteq
\Gamma(\pi).
}$$ Finally note that to ${\mathcal{A}}_{\Delta}$ we can associate the following map $${\overline}{\Sigma}:{\mathbb {R}}^{+}\ni t
\mapsto
\Sigma_{t}
\in
G_{\Delta},$$ for which we have for all $x\in X$ $$\Psi_{{\mathbb {R}}}^{{\mathcal{H}}}({\overline}{\Sigma})
(x)
=
\sigma_{x}.$$

#### Ex3

In the previous example we consider the extreme case in which $\Gamma(\pi)={\mathcal{H}}$. In order to have a model where $\Gamma(\pi)\subset{\mathcal{H}}$ we have to get a more detailed structure, namely the **half-side modular inclusion**. So for any $x\in X$ let ${\left\langle {\mathfrak{N}}_{x}\subset{\mathfrak{E}}_{x},\Omega_{x}{\right\rangle}}$ be a $hsmi^{+}$ and $V_{x}$ the Wiesbrock one-parameter semigroup of unitarities associated to it so $V_{x}\in Hstr({\mathfrak{E}}_{x})^{+}$ such that ${\mathfrak{N}}_{x}=Ad(V_{x}(1)){\mathfrak{E}}_{x}$. Therefore what we are interested in is that for all $t\in{\mathbb {R}}^{+}$ $$\label{20290405}
\begin{cases}
Ad(V_{x}(t))({\mathfrak{E}}_{x})
\subseteq{\mathfrak{E}}_{x},
\\
Ad(V_{x}(t))({\mathfrak{N}}_{x})
\subseteq{\mathfrak{N}}_{x}.
\end{cases}$$ By using the first inclusion in we can set $$\begin{cases}
{\mathcal{A}}_{V}
\doteqdot\{
V_{t}:
X\ni x\mapsto
Ad(V_{x}(t))
{\upharpoonright}{\mathfrak{E}}_{x}
\in 
Aut({\mathfrak{E}}_{x})
\mid t\in{\mathbb {R}}\}
\\
G_{V}\doteqdot
\theta({\mathcal{A}}_{V}){\upharpoonright}{\mathcal{H}}
\\
{\overline}{{\mathcal{V}}}_{t}
\doteqdot
\theta(V_{t}){\upharpoonright}{\mathcal{H}},\,
t\in{\mathbb {R}}.
\end{cases}$$ Hence for all $x\in X$ and $t\in{\mathbb {R}}$ $$\begin{cases}
{\overline}{{\mathcal{V}}}_{t}(v)(x)
=
Ad(V_{x}(t))v(x)
\\
\Psi_{{\mathbb {R}}}^{{\mathcal{H}}}({\overline}{{\mathcal{V}}})
(x)(t)
=
Ad(V_{x}(t))
\end{cases}$$ Therefore if we set ${\mathcal{D}}$ and $\Gamma(\pi)$ such that $${\mathcal{D}}
\subseteq
\Gamma(\pi)
\doteqdot
\int^{\oplus}
{\mathfrak{N}}_{x}\,
d\mu(x)
\subset
{\mathcal{H}}$$ then by using the second inclusion in we have $$\boxed{
{\left\langle G_{V},{\mathcal{D}}{\right\rangle}}
\subseteq
\Gamma(\pi).
}$$

#### Inner property of the Tomita-Takesaki modular group

For any semi-finite von Neumann algebra ${\mathfrak{N}}$ and any $\phi\in{\mathbf{N}}_{{\mathfrak{N}}}$ faithful we have that the **Tomita-Takesaki modular group $\sigma_{{\mathfrak{N}}}^{\phi}$ is inner** (see [@tak2 Thm. $3.14$ Ch. $VIII$]) i.e. it is implemented by a strongly continuous group morphism $V:{\mathbb {R}}\to U({\mathfrak{N}})$, where $U({\mathfrak{N}})\doteqdot\{U^{-1}=U^{*}\mid U\in{\mathfrak{N}}\}$, so in particular $$\label{23452904a}
V({\mathbb {R}})\subset{\mathfrak{N}}.$$ Now let ${\left\langle H_{\phi},\pi_{\phi},\Omega_{\phi}{\right\rangle}}$ be a cyclic representation associated to $\phi$ and ${\mathfrak{N}}_{\phi}\doteqdot\pi({\mathfrak{N}}_{\phi})$ which is a von Neumann algebra $\phi$ being normal, then by immediatedly we have $$\label{23452904b}
\boxed{
\pi_{\phi}(V({\mathbb {R}}))\subset{\mathfrak{N}}_{\phi}.
}$$ By the invariance $\phi=\phi\circ\sigma_{{\mathfrak{N}}}^{\phi}$, and the cited unitary implementation we obtain that there exists $W_{\phi}$ unitary action on $H_{\phi}$ such that $$\label{23372904}
\boxed{
\begin{cases}
Ad(W_{\phi}(t))\circ\pi_{\phi}
=
Ad(\pi_{\phi}(V(t)))\circ\pi_{\phi},
\\
W_{\phi}(t)=\Delta_{\phi}^{it},
\end{cases}
}$$ where the second sentence comes by [@tak2 Thm, $1.2$ Ch. $VIII$], with $\Delta_{\phi}$ the modular operator associated to ${\left\langle {\mathfrak{N}}_{\phi},\Omega_{\phi}{\right\rangle}}$.

 Section of Projections 
========================

In sections \[15491702A\] and \[15491702B\], except when explicitedly stated, we shall maintain Notations \[12042101\].

${\left\langle \nu,\eta,E,Z,T{\right\rangle}}$ invariant set with respect to ${\mathcal{F}}$ {#15491702A}
--------------------------------------------------------------------------------------------

\[19560202\] Let $Z,T$ be two locally compact spaces, $E\in Hlcs$, $\nu\in Rad(Z)$ and $\eta\in Rad(T)^{Z}$. Set $${\mathfrak{L}}_{(1,1)}
(T,E,\eta,\nu)
\doteqdot
\left\{
{\overline}{F}\in\bigcap_{\lambda\in Z}
{\mathfrak{L}}_{1}(T,E,\eta_{\lambda})
\mid
\left(
Z\ni\lambda
\mapsto
\int{\overline}{F}(s)
d\eta_{\lambda}(s)
\in E
\right)
\in
{\mathfrak{L}}_{1}(Z,E,\nu)
\right\}$$

\[19510202\] Let $Z$ be a locally compact space, $\nu\in Rad(Z)$ and $\eta\in Rad(Y)^{Z}$ finally let $D\in\prod_{x\in X}2^{{\mathfrak{E}}_{x}}$ and assume $(A)$ of Lemma \[11011501\]. Thus $(\forall{\overline}{F}\in
{\mathfrak{L}}_{(1,1)}
(Y,{\mathfrak{G}},\eta,\nu))
(\forall x\in X)
(\forall v\in
\prod_{y\in X}D(y))$ $$\Pr_{x}
\circ
\left[
\int
\left(
\int{\overline}{F}(s)\,
d\eta_{\lambda}(s)
\right)\,
d\nu(\lambda)
\right]
(v)
=
\left[
\int
\left(
\int
\Pr_{x}(\Psi({\overline}{F}))(s)
v(x)\,
d\eta_{\lambda}(s)
\right)\,
d\nu(\lambda)
\right].$$

Let ${\overline}{F}\in
{\mathfrak{L}}_{(1,1)}
(Y,{\mathfrak{G}},\eta,\nu)$, $x\in X$ and $v\in\prod_{y\in X}D(y)$. By Theorem \[16291501\] $$\Pr_{x}
\circ
\left[
\int
\left(
\int{\overline}{F}(s)\,
d\eta_{\lambda}(s)
\right)\,
d\nu(\lambda)
\right]
(v)
=
\int
\Pr_{x}\circ\Psi
\left(
\int{\overline}{F}(s)\,d\eta_{(\cdot)}(s)
\right)
(\lambda)
(v(x))
\,d\nu(\lambda).$$ Moreover $\forall\lambda\in Z$ $$\begin{aligned}
{2}
\Pr_{x}\circ\Psi
\left(
\int{\overline}{F}(s)\,d\eta_{(\cdot)}(s)
\right)
(\lambda)
(v(x))
&=
\Pr_{x}
\circ
\left(
\int
{\overline}{F}(s)\,
d\eta_{\lambda}(s)
\right)
\circ
\imath_{x}
(v(x))
\\
&=
\int
\Pr_{x}(\Psi({\overline}{F}))(s)
\circ
\Pr_{x}\circ\imath_{x}
(v(x))
\,d\eta_{\lambda}(s)
\\
&=
\int
\Pr_{x}(\Psi({\overline}{F}))(s)
v(x)
\,d\eta_{\lambda}(s),\end{aligned}$$ where in the first equality we used Prop. \[15111401\], while in the second one Theorem \[16291501\]. Then the statement follows.

\[20030202\] $V$ is a *${\left\langle \nu,\eta,E,Z,T{\right\rangle}}$ invariant set with respect to ${\mathcal{F}}$* if

1.  $T,Z$ are two locally compact spaces;

2.  $E\in Hlcs$ and $V\subseteq E$;

3.  $\nu\in Rad(Z)$ and $\eta\in Rad(T)^{Z}$;

4.  ${\mathcal{F}}
    \subseteq
    {\mathfrak{L}}_{(1,1)}
    (T,E,\eta,\nu)$

5.  $\forall{\overline}{F}\in{\mathcal{F}}$ $$\left[
    \int
    \left(
    \int{\overline}{F}(s)\,
    d\eta_{\lambda}(s)
    \right)\,
    d\nu(\lambda)
    \right]
    V
    \subseteq
    V.$$

\[18310302\] Let us assume the hypotheses of Corollary \[19510202\] and $V$ be a ${\left\langle \nu,\eta,{\mathfrak{G}},Z,Y{\right\rangle}}$ invariant set with respect to ${\mathcal{F}}$ such that $V\cap\prod_{y\in X}D(y)\neq\emptyset$. Then $\forall v\in V\cap\prod_{y\in X}D(y)$ and $\forall{\overline}{F}\in{\mathcal{F}}$ $$\left(
X\ni x
\mapsto
\left[
\int
\left(
\int
\Pr_{x}(\Psi({\overline}{F}))(s)
v(x)\,
d\eta_{\lambda}(s)
\right)\,
d\nu(\lambda)
\right]
\in{\mathfrak{E}}_{x}
\right)
\in
V.$$

By Corollary \[19510202\].

Construction of classes $
\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}
$ through invariant sets  {#15491702B}
---------------------------------------------------------------------------------------------------------------------------

In this Section we shall assume that ${\mathbf{E}}\doteqdot
\{{\mathfrak{E}}_{x}\}_{x\in X}$ is a class of complex Banach spaces. Moreover $
Cld({\mathfrak{E}}_{x})
$ denotes, for all $x\in X$, the class of all closed densely defined linear operators $T_{x}:
Dom(T_{x})
\subseteq
{\mathfrak{E}}_{x}
\to
{\mathfrak{E}}_{x}$.

\[Ch. $9$, $\S1$, $n^{\circ}4$ of [@kato]\] \[13411311biss\] Let $M>1$, $\beta\in{\mathbb {R}}$. Set ${\mathcal{G}}(M,\beta,{\mathbf{E}})$ the class of all $T\in\prod_{x\in X}Cld({\mathfrak{E}}_{x})$ such that $]\beta,\infty[\subseteq 
P(-T(x))$ [^17] and $(\forall\xi>\beta)
(\forall k\in{\mathbb{N}})
(\forall x\in X)$ $$\|(T(x)+\xi)^{-k}\|_{B({\mathfrak{E}}_{x})}
\leq
M
(\xi-\beta)^{-k}.$$ Moreover let us denote by $
\{e^{-tT(x)}\}_{t\in{\mathbb {R}}^{+}}
$ the strongly continuous semigroup generated by $-T(x)$.

\[ Separation of the Spectrum \] \[13361311biss\] See [@kato $n^{\circ}4$,$\S 6$, Ch. $3$]. Let $M>1$, $\beta\in{\mathbb {R}}$. We say that $T\in
{\mathcal{G}}(M,\beta,{\mathbf{E}})$ satisfies the property of separation of the spectrum if $(\exists\,\Gamma)
(\forall x\in X)
(\exists\,\Sigma_{T(x)}'
\subseteq\Sigma(T(x)))
(\exists\,A_{T(x)}\in Op({\mathbb {C}}))$ such that $\Gamma$ is a closed curve in ${\mathbb {C}}$, $\Sigma_{T(x)}'$ is bounded and $$\Sigma_{T(x)}'
\subset
A_{T(x)}
\subset
{\mathcal{O}}_{i}(\Gamma),
\,
\Sigma_{T(x)}''
\subset
{\mathcal{O}}_{e}(\Gamma).$$ Here ${\mathcal{O}}_{i}(\Gamma)$ is the interior of $\Gamma$, namely the compact set of ${\mathbb {C}}$ whose frontier is $\Gamma$, ${\mathcal{O}}_{e}(\Gamma)
\doteqdot
\complement
{\mathcal{O}}_{i}(\Gamma)$ is the exterior of $\Gamma$, $\Sigma(T(x))$ is the spectrum of $T(x)$, finally $\Sigma_{T(x)}''
\doteqdot
\Sigma(T(x))
\cap
\complement
\Sigma_{T(x)}'$.

Let $T\in{\mathcal{G}}(M,\beta,{\mathbf{E}})$ satisfy the property of separation of the spectrum, then $\forall x\in X$ we set $$\label{16171102}
P(x)
\doteqdot
-
\frac{1}{2\pi i}
\int_{\Gamma}
R(T(x);\zeta)\,
d\zeta
\in B({\mathfrak{E}}_{x}).$$ Moreover set $R_{T}^{\rho}\in\prod_{x\in X}
{\mathcal{L}}({\mathfrak{E}}_{x})^{{\mathbb {R}}^{+}}$ such that $R_{T}^{\rho}
(x)(s)\doteqdot R(T(x);\rho(s))$, for all $x\in X$ and $s\in K(\Gamma)$, while $R_{T}^{\rho}(x)(s)\doteqdot{\mathbf{0}}$, if $s\in{\mathbb {R}}^{+}-K(\Gamma)$. Here $R(T(x);\cdot):
P(T(x))
\ni \zeta
\mapsto
(T(x)-\zeta)^{-1}
\in B({\mathfrak{E}}_{x})
$ is the resolvent map of $T(x)$ and $P(T(x))$ is its resolvent set, while the integration is with respect to the norm topology on $B({\mathfrak{E}}_{x})$.

\[15221311biss\] Let $M>1$, $\beta\in{\mathbb {R}}$ and $T\in
{\mathcal{G}}(M,\beta,{\mathbf{E}})$ satisfy the property of separation of the spectrum. Then for all $x\in X$ by [@kato Th. $6.17.$, Ch. $3$], $P(x)\in\Pr({\mathfrak{E}}_{x})$ and ${\mathfrak{E}}_{x}=
M_{x}'
\oplus
M_{x}''$ direct sum of two closed subspaces of ${\mathfrak{E}}_{x}$, where $M_{x}'=P(x){\mathfrak{E}}_{x}$ and $M_{x}''
=
({\mathbf{1}}_{x}-P(x))
{\mathfrak{E}}_{x}$. Moreover $T(x)$ decomposes according the previous decomposition, namely $T(x){\upharpoonright}M_{x}'
\in B(M_{x}')$ such that $\Sigma(T_{x}{\upharpoonright}M_{x}')
=\Sigma_{T_{x}}'$ and $T_{x}{\upharpoonright}M_{x}''$ is a closed operator in $M_{x}''$ such that $\Sigma(T_{x}{\upharpoonright}M_{x}'')=\Sigma_{T_{x}}''$.

\[16420402\] Let $K(\Gamma)
\subset{\mathbb {R}}^{+}$ a compact set, $A$ an open neighbourhood of $K(\Gamma)$ and $\rho:A\to{\mathbb {C}}$ be such that $\rho\in C_{1}(A,{\mathbb {R}}^{2})$ [^18] and $
\rho(K(\Gamma))
=
\Gamma 
$. Set $\forall s\in K(\Gamma)$, $\eta_{s}\in Radon({\mathbb {R}}^{+})$ such that $$\eta_{s}:
{\mathcal{C}_{cs}
\left({\mathbb {R}}^{+}\right)}\ni f
\mapsto
\int_{{\mathbb {R}}^{+}}e^{\rho(s)t}f(t)\,dt.$$ Moreover let $\nu\in Radon({\mathbb {R}}^{+})$ be the ${\mathbf{0}}-$extension of $\nu_{0}\in Radon(K(\Gamma))$ such that $$\nu_{0}:{\mathcal{C}_{cs}
\left(K(\Gamma)\right)}
\ni g
\mapsto
\int_{K(\Gamma)}
\frac{- g(s)}{2\pi i}
\frac{d\rho}{ds}(s)\,
ds.$$ Finally let $M>1$, $\beta\in{\mathbb {R}}$ and $T\in{\mathcal{G}}(M,\beta,{\mathbf{E}})$, then we set ${\mathcal{W}}_{T}
\in
\prod_{x\in X}{\mathbf{U}}(B_{s}({\mathfrak{E}}_{x}))$ such that $(\forall x\in X)(\forall t\in{\mathbb {R}}^{+})$ $$\begin{cases}
{\mathcal{W}}_{T}(x)(t)
\doteqdot
e^{-T(x)t},\\
{\overline}{F}_{T}\doteqdot\Lambda({\mathcal{W}}_{T}),
\end{cases}$$ where $\Lambda$ has been defined in Def. \[14321401\].

\[15251311biss\] Let $M>1$, $\beta\in{\mathbb {R}}$ and $T\in{\mathcal{G}}(M,\beta,{\mathbf{E}})$ satisfy the property of separation of the spectrum. Assume that there exists a closed curve $\Gamma$ of which in Definition \[13361311biss\] such that $$\label{19281311biss}
Re(\Gamma)
\subseteq
{\mathbb {R}}^{-}.$$ Then $\forall x\in X$ and $\forall v_{x}\in{\mathfrak{E}}_{x}$ $$\label{17551311biss}
P(x)v_{x}
=
-
\frac{1}{2\pi i}
\int_{K(\Gamma)}
\frac{d\rho}{ds}(s)
R(T(x);\rho(s))v_{x}
\,ds,$$ and $\forall s\in K(\Gamma)$, $$\label{18150402}
R(T(x),\rho(s))v_{x}
=
\int_{0}^{\infty}
e^{\rho(s) t}
e^{-tT(x)}
v_{x}\,dt
=
\int_{{\mathbb {R}}^{+}}
{\mathcal{W}}_{T}(x)(t)v_{x}\,
d\eta_{s}(t).$$ Here the integration is with respect to the norm topology on ${\mathfrak{E}}_{x}$. Moreover let ${\mathfrak{G}}$ be that in Def. \[10221801\] relative to ${\mathbf{E}}$. If ${\overline}{F}_{T}\in
{\mathfrak{L}}_{(1,1)}
({\mathbb {R}}^{+},{\mathfrak{G}},\eta,\nu)$ and $V$ is a ${\left\langle \nu,\eta,{\mathfrak{G}},K(\Gamma),{\mathbb {R}}^{+}{\right\rangle}}$ invariant set with respect to $\{{\overline}{F}_{T}\}$, then [^19] $$\label{15351311biss}
P\bullet V\subseteq V
$$

By , [@IntBourb $IV.35$ Th. $1$], and by the norm continuity of the map $
B({\mathfrak{E}}_{x})\ni A\mapsto Aw\in
{\mathfrak{E}}_{x}$ for any $w\in{\mathfrak{E}}_{x}$, we have . Moreover by we can apply [@kato eq. $1.28$, $n^{\circ}3$, $\S 1$, Ch. $9$] and follows by Def. \[16420402\]. Fix $v\in V$ so $\forall x\in X$ $$\begin{aligned}
{1}
\label{18590402}
P(x)v(x)
&=
-
\frac{1}{2\pi i}
\int_{K(\Gamma)}
\frac{d\rho}{ds}(s)
R(T(x);\rho(s))v(x)
\,ds
\notag
\\
&=
-
\frac{1}{2\pi i}
\int_{K(\Gamma)}
\frac{d\rho}{ds}(s)
\left(
\int_{{\mathbb {R}}}
{\mathcal{W}}_{T}(x)(t)v(x)\,
d\eta_{s}(t)
\right)
\,ds
\notag
\\
&=\int_{K(\Gamma)}
\left(
\int_{{\mathbb {R}}}
\Pr_{x}
\left(
\Psi({\overline}{F}_{T})
\right)(t)
v(x)\,
d\eta_{s}(t)\right)\,
d\nu(s).\end{aligned}$$ Here the first equality comes by , the second one by and the third one by Prop. \[15111401\] and Def. \[16420402\]. Finally with the notations in Corollary \[15111901\] we have that for the strong operator topology we can choose $(\forall x\in X)
(S_{x}={\mathcal{P}}_{\omega}({\mathfrak{E}}_{x}))$, thus $D(x)={\mathfrak{E}}_{x}$, for all $x\in X$, and by Corollary \[15111901\] holds $(A)$ of Lemma \[11011501\]. Therefore the statement follows by and Proposition \[18310302\].

\[13511102\] Let us assume the hypotheses and notations of the Main Theorem \[17301812b\], and the hypotheses of Lemma \[15251311biss\] where $T$ is such that $-T(x)$ is the infinitesimal generator of ${\mathcal{U}}(x)$, for all $x\in X$. Moreover let $V\subset\prod_{x\in X}{\mathfrak{E}}_{x}$. Finally let ${\mathfrak{D}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{B}},\gamma{\right\rangle}},\eta,X,{\mathfrak{L}}{\right\rangle}}$ and ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that $$\label{15181102}
\Gamma(\eta)
\subseteq
\left\{
F\in\prod_{x\in X}{\mathcal{L}}({\mathfrak{E}}_{x})
\mid
F\bullet V
\subseteq V
\right\},$$ ${\overline}{F}_{T}\in
{\mathfrak{L}}_{(1,1)}
({\mathbb {R}}^{+},{\mathfrak{G}},\eta,\nu)$ and $V$ is a ${\left\langle \nu,\eta,{\mathfrak{G}},K(\Gamma),{\mathbb {R}}^{+}{\right\rangle}}$ invariant set with respect to $\{{\overline}{F}_{T}\}$, $\{{\mathcal{U}}\}
\in
\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$.

Notice that ${\mathcal{E}}(\Theta)={\mathcal{E}}$ so if ${\mathcal{E}}
\subseteq
V
\subseteq
\Gamma(\pi)$ then implies that the $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ is compatible.

By and by and .

\[18421202\] Our aim now is to see when $V$ is a ${\left\langle \nu,\eta,{\mathfrak{G}},K(\Gamma),{\mathbb {R}}^{+}{\right\rangle}}$ invariant set with respect to $\{{\overline}{F}_{T}\}$, by maintaining the positions ${\mathcal{E}}
\subseteq
V
\subseteq
\Gamma(\pi)$ and , ensuring as remarked, that ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ is compatible. By using Corollary \[28111707\], we can try to show that if $V$ is a ${\left\langle \nu,\eta,{\mathfrak{G}},K(\Gamma),{\mathbb {R}}^{+}{\right\rangle}}$ invariant set with respect to $\Lambda(\Gamma(\rho))$ then $V$ is a ${\left\langle \nu,\eta,{\mathfrak{G}},K(\Gamma),{\mathbb {R}}^{+}{\right\rangle}}$ invariant set with respect to $\Lambda(\Gamma^{x_{\infty}}(\rho))$, hence by , with respect to $\{{\overline}{F}_{T}\}$. Might be at first glance not a weakening, but it is indeed the most powerfull way for constructing bundles of $\Omega-$spaces, namely that described in Def. \[17471910Ba\], allows us to choose, when $X$ is compact, the set $\Gamma(\rho)$, see Remark \[17150312\].

Assume the notations in Def. \[20030202\]. Another way, maybe better than the previous one, is the following. Let $v\in\prod_{x\in X}{\mathfrak{E}}_{x}$ and $A$ be a subalgebra of ${\mathfrak{G}}$, thus if we set $$V\doteqdot
{\overline}{\textrm{span}}(Av),$$ closure in $\prod_{x\in X}{\mathfrak{E}}_{x}$, then by Lemma \[17141401\] we obtain $$A V\subseteq V.$$ Thus if ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}$ is a topological algebra for all $x\in X$, it is sufficient to take for example the closure in ${\mathfrak{G}}$ of the algebra $A_{0}$ [^20] generated by any subset of ${\mathfrak{G}}$ which contains the set $$\left\{
\int
\left(
\int{\overline}{F}(s)\,
d\eta_{\lambda}(s)
\right)\,
d\nu(\lambda)
\mid
{\overline}{F}\in{\mathcal{F}}
\right\}.$$ Notice that by the fact that $P(x)\in\Pr({\mathfrak{E}}_{x})$ for all $x\in X$, we deduce by and Corollary \[19510202\] that $$\label{20081102}
a_{T}
\doteqdot
\int
\left(
\int{\overline}{F}_{T}(s)\,
d\eta_{\lambda}(s)
\right)\,
d\nu(\lambda)
\in\Pr
\left(\prod_{x\in X}{\mathfrak{E}}_{x}\right).$$ Thus the algebra $A_{0}$ should be generated by any subset of ${\mathfrak{G}}$ which contains not only the operator $a_{T}$, otherwise $A_{0}={\mathbb{K}}\cdot a_{T}$ and then $A={\mathbb{K}}\cdot a_{T}$ which is not interesting. The last equality comes by the fact that ${\mathbb{K}}\cdot v$ is a closed set for any $v\in Z-\{{\mathbf{0}}\}$ where $Z$ is a topological vector space such that that $Z^{*}$ separates the points of $Z$, for example any Hlcs. Indeed let $v,w\in Z$ such that $v\neq{\mathbf{0}}$ and $\{\lambda_{\alpha}\}_{\alpha\in D}$ a net in $Z$ such that $\lim_{\alpha\in D}\lambda_{\alpha}v=w$. Thus there exists $\phi\in Z^{*}$ such that $\phi(v)\neq 0$ and $\{\lambda_{\alpha}\phi(v)\}_{\alpha\in D}$ is a Cauchy net in ${\mathbb{K}}$. But $\phi(v)\neq 0$ so also $\{\lambda_{\alpha}\}_{\alpha\in D}$ is a Cauchy net in ${\mathbb{K}}$, let $\mu=\lim_{\alpha\in D}\lambda_{\alpha}$. Thus $w=\lim_{\alpha\in D}\lambda_{\alpha}v
=\mu v$, which show that ${\mathbb{K}}v$ is closed in $Z$.

Construction of classes $
\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}
$ through the generalized Lebesgue Theorem  {#15511702}
---------------------------------------------------------------------------------------------------------------------------

In this section $X$ is a topological space, $Y$ is a locally compact space $\mu$ is a Radon measure on $Y$, and ${\mathfrak{V}}
=
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ is a bundle of $\Omega-$spaces, we indicate with $
{\mathfrak{N}}\doteqdot
\{
\nu_{j}\mid
j\in J
\}
$ the directed set of seminorms on ${\mathfrak{E}}$.

\[22132802\] Let $Z\in Hlcs$ and $\{\psi_{i}\mid i\in I\}$ a fundamental set of seminorms on $Z$. We denote by $$\left(Z^{Y}\right)_{s}$$ the $Hlcs$ whose linear space support is $Z^{Y}$ and whose $Hlct$ is that generated by the following set of seminorms $$\label{20531902}
\begin{cases}
\{q_{s}^{i}\mid s\in Y, i\in I\},
\\
q_{s}^{i}:
Z^{Y}
\ni 
f
\mapsto
\psi_{i}(f(s)).
\end{cases}$$ Moreover for any $B\subseteq Z^{Y}$ we shall denote by $B_{s}$ the $Hlc$ subspace of $\left(Z^{Y}\right)_{s}$. Notice that this definition is well-set being indipendent by the choice of the fundamental set of seminorms, indeed the topology is that of uniform convergence over the finite subsets of $Y$.

\[15361702\] Set $$\begin{cases}
\overset{\mu}{\blacklozenge}:
\prod_{x\in X}
{\mathfrak{L}}_{1}(Y,{\mathfrak{E}}_{x},\mu)
\to
\prod_{x\in X}
{\mathfrak{E}}_{x},
\\
\overset{\mu}{\blacklozenge}(H)(x)
\doteqdot
\int H(x)(s)\,
d\mu(s)
\in
{\mathfrak{E}}_{x},
\end{cases}$$ for all $H\in
\prod_{x\in X}
{\mathfrak{L}}_{1}(Y,{\mathfrak{E}}_{x},\mu)$ and for all $x\in X$

\[12121902\] Set $$\begin{cases}
\bigstar:
\prod_{x\in X}
{\mathcal{L}}({\mathfrak{E}}_{x})^{Y}
\times
\prod_{x\in X}
{\mathfrak{E}}_{x}
\to
\prod_{x\in X}{\mathfrak{E}}_{x}^{Y},
\\
(\forall x\in X)(\forall s\in Y)
(F\bigstar v)(x)(s)
\doteqdot
F(x)(s)(v(x)).
\end{cases}$$

\[15492502\] *${\left\langle {\mathfrak{V}},{\mathfrak{Z}}{\right\rangle}}$ are $\mu-$related* if

1.  ${\mathfrak{Z}}
    \doteqdot
    {\left\langle {\left\langle {\mathfrak{T}},\gamma{\right\rangle}},\zeta,X,{\mathfrak{K}}{\right\rangle}}$ be a bundle of $\Omega-$spaces;

2.  for all $x\in X$ [^21] $$\label{20242906}
    \begin{cases}
    {\mathfrak{T}}_{x}
    \subseteq
    Meas(Y,{\mathfrak{E}}_{x},\mu)
    \bigcap
    {\mathfrak{L}}_{1}(Y,{\mathfrak{E}}_{x},\mu)_{s},
    \\
    {\mathfrak{K}}_{x}
    =
    \left\{
    \sup_{(s,j)\in O}
    q_{(s,j)}^{x}
    \mid
    O\in{\mathcal{P}}_{\omega}(Y\times J)
    \right\},
    \\
    q_{(s,j)}^{x}:
    {\mathfrak{T}}_{x}
    \ni
    f_{x}\mapsto
    \nu_{j}(f_{x}(s)),
    \forall
    s\in Y,
    j\in J;
    \end{cases}$$

3.  $\Gamma(\zeta)
    \subset
    \left[
    \prod_{x\in X}^{b}
    {\mathfrak{T}}_{x}
    \right]_{ui}$;

4.  $\overset{\mu}{\blacklozenge}(\Gamma(\zeta))
    \subseteq
    \Gamma(\pi)$.

Here we set for all ${\mathcal{A}}\subseteq
\prod_{x\in X}^{b}
{\mathfrak{T}}_{x}$ $$\label{17180207}
\left[
{\mathcal{A}}\right]_{ui}
\doteqdot
\left\{
H\in{\mathcal{A}}\mid
(\forall j\in J)
\left(
\int_{Y}^{\bullet}
\sup_{x\in X}
\nu_{j}(H(x)(s))
\,d|\mu|(s)
<\infty
\right)
\right\}$$ Finally *${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{H}}{\right\rangle}}$ are $\mu-$related* if

1.  ${\mathbf{H}}=\{{\mathbf{H}}_{x}\}_{x\in X}$ such that ${\mathbf{H}}_{x}\subseteq
    {\mathcal{L}}({\mathfrak{E}}_{x})^{Y}$ for all $x\in X$,

2.  ${\left\langle {\mathfrak{V}},{\mathfrak{Z}}{\right\rangle}}$ are $\mu-$related

3.  $$\label{19090907}
    \left(
    \prod_{x\in X}{\mathbf{H}}_{x}\right)
    \bigstar
    \left(
    \prod_{x\in X}{\mathfrak{E}}_{x}\right)
    \subseteq
    \prod_{x\in X}{\mathfrak{T}}_{x}.$$

\[**GLT**\] \[15101701\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{Z}}{\right\rangle}}$ be $\mu-$related. Then for all $x\in X$ $$\overset{\mu}{\blacklozenge}
\left(
\left[\Gamma_{\diamond}^{x}(\zeta)
\right]_{ui}
\right)
\subseteq
\Gamma_{\diamond}^{x}(\pi).$$

Let $x\in X$ and $F\in\left[\Gamma_{\diamond}^{x}
(\zeta)\right]_{ui}$ thus by Corollary \[28111707\] there exists $\eta\in\Gamma(\zeta)$ such that for all $j\in J,s\in Y$ $$\label{12221702}
\begin{cases}
F(x)=\eta(x)
\\
\lim_{z\to x}
\nu_{j}(F(z)(s)-\eta(z)(s))
=0.
\end{cases}$$ Fix $j\in J$ thus by [@IntBourb Prop.$6$, $No 2$, $\S 1$, Ch. $6$] for all $z\in X$ $$\label{14550907}
\nu_{j}
\left(
\int
(F(z)(s)-\eta(z)(s))
\,d\mu(s)
\right)
\leq
\int^{\bullet}
\nu_{j}(F(z)(s)-\eta(z)(s))\,
d|\mu|(s)$$ Moreover $\nu_{j}^{z}$ is continuous by definition of bundles of $\Omega-$spaces, while $F(z)$ and $\eta(z)$ are by construction $\mu-$measurable, hence by [@IntBourb Thm. $1$; Cor. $3$, $n^{\circ}3$, $\S\,5$,Ch. $4$] the map $Y\ni s\mapsto
\nu_{j}(F(z)(s)-\eta(z)(s))$ is $\mu-$measurable thus $|\mu|-$measurable. Moreover by the hypothesis on $F$ and by $(3)$ of Def. \[15492502\] $$\label{15370907}
\int^{\bullet}
\nu_{j}(F(z)(s)-\eta(z)(s))\,
d|\mu|(s)
\leq
\int^{\bullet}
\left(
\sup_{x\in X}
\nu_{j}(F(x)(s))
+
\sup_{x\in X}
\nu_{j}(\eta(x)(s))
\right)
d|\mu|(s)
<\infty.$$ Therefore by [@IntBourb Prp..$9$, $No 3$, $\S 1$, Ch. $5$] the map $Y\ni s\mapsto
\nu_{j}(F(z)(s)-\eta(z)(s))$ is $|\mu|-$ essentially integrable hence by the fact that $\int_{Y}^{\bullet}f\,d|\mu|
=\int_{Y}f\,d|\mu|$ for all $|\mu|-$essentially integrable map $f$, we have by $$\label{12541702}
\nu_{j}
\left(
\int
(F(z)(s)-\eta(z)(s))
\,d\mu(s)
\right)
\leq
\int
\nu_{j}(F(z)(s)-\eta(z)(s))
\,
d|\mu|(s).$$ Let $\{z_{n}\}_{n}\subset X$ be such that $\lim_{n\in{\mathbb{N}}}z_{n}=x$ thus by $$\label{12491702A}
\lim_{n\in{\mathbb{N}}}
\nu_{j}(F(z_{n})(s)-\eta(z_{n})(s))
=0,$$ For all $s\in Y$ $$\nu_{j}(F(z)(s)-\eta(z)(s))\,
\leq
\sup_{x\in X}
\nu_{j}(F(x)(s))
+
\sup_{x\in X}
\nu_{j}(\eta(x)(s))$$ thus by the hypothesis on $F$, by $(3)$ of Def. \[15492502\], by the fact that $\int_{Y}^{\bullet}\leq\int_{Y}^{*}$, by , and by the Lebesgue Theorem [@IntBourb Th.$6$, $No 7$, $\S 3$, Ch. $4$] we have $$\label{12531702}
\lim_{n\in{\mathbb{N}}}
\int
\nu_{j}(F(z_{n})(s)-\eta(z_{n})(s))\,
d|\mu|(s)
=0.$$ Finally by , and the fact that $X$ is metrizable we obtain $$\lim_{z\to x}
\nu_{j}
\left(
\int
F(z)(s)
\,
d\mu(s)
-
\int
\eta(z)(s)
\,
d \mu(s)
\right)
=0,$$ thus the statement follows by hypothesis $(4)$ and Corollary \[28111707\].

\[ **$\left(\Theta,{\mathcal{E}},\mu\right)-$structures**\] \[17161902\] We say that ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ is a *$\left(\Theta,{\mathcal{E}},\mu\right)-$ structure* if

1.  $
    {\mathfrak{V}}
    \doteqdot
    {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}
    $ is a bundle of $\Omega-$spaces;

2.  ${\mathcal{E}}\subseteq\Gamma(\pi)$;

3.  $\Theta\subseteq
    \prod_{x\in X}
    Bounded({\mathfrak{E}}_{x})$;

4.  $\forall B\in\Theta$

    1.  ${\mathbf{D}}(B,{\mathcal{E}})
        \ne\emptyset$;

    2.  $\bigcup_{B\in\Theta}{\mathcal{B}}_{B}^{x}$ is total in ${\mathfrak{E}}_{x}$ for all $x\in X$;

5.  ${\mathfrak{Q}}
    \doteqdot
    {\left\langle {\left\langle {\mathfrak{H}},\gamma{\right\rangle}},\xi,X,{\mathfrak{Y}}{\right\rangle}}$ is a bundle of $\Omega-$spaces such that for all $x\in X$ $$\label{22512906}
    \boxed{
    \begin{cases}
    {\mathfrak{H}}_{x}
    \subseteq
    {\mathfrak{L}}_{1}
    \left(Y,{\mathcal{L}}_{S_{x}}
    ({\mathfrak{E}}_{x}),\mu\right)_{s},
    \\
    {\mathfrak{Y}}_{x}
    =
    \left\{
    \sup_{(t,j,B)\in O}
    P_{(t,j,B)}^{x}
    \mid
    O\in{\mathcal{P}}_{\omega}
    \left(Y\times J\times\Theta\right)
    \right\}
    \\
    P_{(t,j,B)}^{x}:
    {\mathfrak{H}}_{x}
    \ni
    F\mapsto
    \sup_{v\in{\mathbf{D}}(B,{\mathcal{E}})}
    \nu_{j}(F(t)v(x)),\,
    \forall
    t\in Y,
    B\in\Theta,
    j\in J.
    \end{cases}
    }$$

Here $S_{x}$, ${\mathcal{B}}_{B}^{x}$ and ${\mathbf{D}}(B,{\mathcal{E}})$ are defined in . Moreover ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ is an *invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure* if it is a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure such that $$\label{18112502}
\left\{
F\in\prod_{z\in X}^{b}
{\mathfrak{H}}_{z}
\mid
(\forall t\in Y)
(F_{t}
\bullet
{\mathcal{E}}(\Theta)
\subseteq
\Gamma(\pi))
\right\}
=
\Gamma(\xi).$$

\[14350303\] Let $\mu_{\lambda}$ for all $\lambda>0$ be defined as in Def. \[15062301\], let ${\mathfrak{Q}}
=
{\left\langle {\left\langle {\mathfrak{H}},\gamma{\right\rangle}},\xi,X,{\mathfrak{S}}{\right\rangle}}$, ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure, $x\in X$, ${\mathcal{O}}\subseteq\Gamma(\xi)$. and ${\mathcal{D}}\subseteq\Gamma(\pi)$. Set $${\mathbf{Lap}}({\mathfrak{V}})(x)
\doteqdot
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},
{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\mu_{\lambda}).$$ Assume that $$\label{12480707}
\Gamma_{{\mathcal{O}}}^{x}(\xi)
\bigcap
{\mathbf{Lap}}({\mathfrak{V}})(x)
\neq
\emptyset$$ We say that ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the *weak-Laplace duality property on ${\mathcal{O}}$ and ${\mathcal{D}}$ at $x$*, shortly $w-{\mathbf{LD}}_{x}({\mathcal{O}},{\mathcal{D}})$ if $\forall\lambda>0$ $$\blacksquare_{\mu_{\lambda}}
\left(
\Gamma_{{\mathcal{O}}}^{x}(\xi)
\bigcap
{\mathbf{Lap}}({\mathfrak{V}})(x),
\Gamma_{{\mathcal{D}}}^{x}(\pi)
\right)
\subseteq
\Gamma^{x}(\pi).$$

\[18072802\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure and denote ${\mathfrak{W}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$. Assume that for all $x\in X$ $$\label{22372906}
{\mathfrak{M}}_{x}
\subseteq
{\mathfrak{L}}_{1}
\left(Y,{\mathcal{L}}_{S_{x}}
({\mathfrak{E}}_{x}),\mu\right).$$ Set ${\mathbf{M}}^{\mu}
\doteqdot
\{{\left\langle {\mathbf{M}}_{x}^{\mu},{\mathfrak{Y}}_{x}{\right\rangle}}\}_{x\in X}$ where for all $x\in X$ $$\begin{cases}
{\mathbf{M}}_{x}^{\mu}
\doteqdot
{\overline}{
\left\{
\sigma(x)
\mid
\sigma\in
\Gamma(\rho)
\right\}
},
\\
{\mathfrak{Y}}_{x}
\doteqdot
\{
\sup_{(t,j,B)\in O}
P_{(t,j,B)}^{x}
\mid
O\in{\mathcal{P}}_{\omega}(Y\times J
\times\Theta)
\},
\end{cases}$$ where the closure is in the space $${\mathfrak{L}}_{1}
\left(Y,{\mathcal{L}}_{S_{x}}
({\mathfrak{E}}_{x}),\mu\right)_{s},$$ and ${\mathbf{M}}_{x}^{\mu}$ has to be considered as Hlc subspace of it, finally here $P_{(t,j,B)}^{x}$ is defined on ${\mathbf{M}}_{x}^{\mu}$ as in . Notice that ${\mathbf{M}}^{\mu}$ is a nice family of Hlcs, and that $\Gamma(\rho)$ satisfies by construction $FM(3)$ with respect to ${\mathbf{M}}^{\mu}$. Moreover by and the fact that $\{t\}\in Comp(Y)$ for all $t\in Y$ $$\label{22582906}
P_{(t,j,B)}^{x}
=q_{(\{t\},j,B)}^{x}.$$ By [@gie Cor.$1.6.(iii)$] we deduce that $\Gamma(\rho)$ satisfies $FM(4)$ with respect to $\{{\left\langle {\mathfrak{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}}\}_{x\in X}$. Therefore we obtain by and that for all $t\in Y$, $j\in J$, $B\in\Theta$ and for all $\sigma\in\Gamma(\rho)$ $$X\ni x\mapsto P_{(t,j,B)}^{x}(\sigma(x))
\text{ is $u.s.c.$}.$$ Moreover the upper envelope of a finite set of $u.s.c.$ maps is an $u.s.c.$ map, see [@BourGT Thm.$4$,$\S 6.2.$,Ch.$4$], therefore for all $O\in{\mathcal{P}}_{\omega}(Y\times J
\times\Theta)$ $$\label{16392906}
X\ni x\mapsto 
\sup_{(t,j,B)\in O}
P_{(t,j,B)}^{x}(\sigma(x))
\text{ is $u.s.c.$}.$$ Hence $\Gamma(\rho)$ satisfies $FM(4)$ with respect to ${\mathbf{M}}^{\mu}$. Finally by the boundedness of $\Gamma(\rho)$ by definition and by we have also that for all $\sigma\in\Gamma(\rho)$ and $O\in{\mathcal{P}}_{\omega}(Compact(Y)\times J
\times\Theta)$ $$\sup_{x\in X}
\sup_{(t,j,B)\in O}
P_{(t,j,B)}^{x}(\sigma(x))
<\infty.$$ Therefore we can construct the bundle generated by the couple ${\left\langle {\mathbf{M}}^{\mu},\Gamma(\rho){\right\rangle}}$, see Def. \[17471910Ba\] $${\mathfrak{V}}({\mathbf{M}}^{\mu},\Gamma(\rho))$$ We shall call ${\left\langle {\mathfrak{V}},{\mathfrak{V}}({\mathbf{M}}^{\mu},
\Gamma(\rho)),X,Y{\right\rangle}}$ the *$\left(\Theta,{\mathcal{E}},\mu\right)-$ structure underlying ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$*.

\[19421902\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure and $A\subset\prod_{x\in X}{\mathfrak{H}}_{x}$. Define $A_{peq}$ as the set of all pointwise equicontinuous elements in $A$, and $A_{ceq}$ as the set of all compactly equicontinuous elements in $A$, see Def. \[21031238\].

\[23222906\] Lemma \[15482712\] holds by replacing a $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ with a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ and $K\in Comp(Y)$ with $t\in Y$. In what follows when referring to Lemma \[15482712\] for a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure we shall mean the corresponding result with the replacements described here.

\[12151902\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure and ${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{H}}{\right\rangle}}$ be $\mu-$related, where ${\mathfrak{Q}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{H}},\gamma{\right\rangle}},\xi,X,{\mathfrak{Y}}{\right\rangle}}$ and ${\mathbf{H}}_{x}\doteqdot{\mathfrak{H}}_{x}$ for all $x\in X$. Thus $$\Gamma(\xi)
\bigstar
{\mathcal{E}}(\Theta)
\subseteq
\Gamma(\zeta)
\Rightarrow
(\forall x\in X)
\left(
\Gamma_{\diamond}^{x}(\xi)_{peq}
\bigstar
\Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi)
\subseteq
\Gamma_{\diamond}^{x}(\zeta)\right).$$

Let $j\in J$, $x\in X$ and $w\in\Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi)$, so there exists $v\in{\mathcal{E}}(\Theta)$ such that $v(x)=w(x)$ then by Cor. \[28111707\] $$\label{14311902}
\lim_{z\to x}
\nu_{j}(w(z)-v(z))
=0.$$ Moreover let $F\in\Gamma_{\diamond}^{x}(\xi)$, so by Lemma \[15482712\] $\exists\,\sigma\in\Gamma(\xi)$ such that $F(x)=\sigma(x)$ and for all $t\in Y$ $$\label{14301902}
\lim_{z\to x}
\nu_{j}\left(F(z)(t)v(z)-\sigma(z)(t)v(z)
\right)=0.$$ Moreover $(\forall t\in Y)
(\exists\,M_{(t,j)}>0)
(\exists\,j_{1}\in J)
(\forall z\in X)$ $$\begin{aligned}
{1}
\nu_{j}\left((F\bigstar w)(z)(t)-
(\sigma\bigstar v)(z)(t)\right)
&=
\nu_{j}\left(F(z)(t)w(z)-
\sigma(z)(t)v(z)\right)
\\
&\leq
\nu_{j}\left(F(z)(t)(w(z)-v(z))\right)
+
\nu_{j}\left(F(z)(t)v(z)-\sigma(z)(t)v(z)
\right)
\\
&\leq
M_{(t,j)}
\nu_{j_{1}}(w(z)-v(z))
+
\nu_{j}\left(F(z)(t)v(z)-\sigma(z)(t)v(z)
\right).\end{aligned}$$ Therefore by and for all $t\in Y$ $$\lim_{z\to x}
\nu_{j}\left((F\bigstar w)(z)(t)-
(\sigma\bigstar v)(z)(t)\right)
=0.$$ Moreover $(\forall t\in Y)
(\exists\,M_{(t,j)}>0)
(\exists\,j_{1}\in J)$ $$\label{18590907}
\sup_{z\in X}
\nu_{j}((F\bigstar w)(z)(t))
\leq
M_{(t,j)}
\sup_{z\in X}
\nu_{j_{1}}(w(z))
<\infty.$$ By the antecedent of the implication of the statement we deduce that $\sigma\bigstar v\in\Gamma(\zeta)$ hence the statement follows by Cor. \[28111707\], , by the fact that by $F\bigstar w\in\prod_{x\in X}{\mathfrak{T}}_{x}$ and by .

\[17441202bis\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a compatible $\left(\Theta,{\mathcal{E}}\right)-$structure. Then for all $x\in X$ $$(\Gamma_{\diamond}^{x}(\rho)_{peq})_{t}
\bullet 
\Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi)
\subseteq
\Gamma_{\diamond}^{x}(\pi)$$

Notice that $(F\bigstar v)(t)=F(\cdot)(t)\bullet v$ with obvious meaning of the symbols. Thus if we set $Y=\{pt\}$ the statement follows by Lm. \[12151902\].

\[15262502\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure and ${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{H}}{\right\rangle}}$ be $\mu-$related, Thus for all $x\in X$ $$\Gamma(\xi)
\bigstar
{\mathcal{E}}(\Theta)
\subseteq
\Gamma(\zeta)
\Rightarrow
\overset{\mu}{\blacklozenge}
\left(
\left[
\Gamma_{\diamond}^{x}(\xi)_{peq}
\bigstar
\Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi)
\right]_{ui}
\right)
\subseteq
\Gamma_{\diamond}^{x}(\pi).$$ Here ${\mathfrak{Q}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{H}},\gamma{\right\rangle}},\xi,X,{\mathfrak{Y}}{\right\rangle}}$, ${\mathfrak{Z}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{T}},\delta{\right\rangle}},\zeta,X,{\mathfrak{K}}{\right\rangle}}$ and ${\mathbf{H}}_{x}\doteqdot{\mathfrak{H}}_{x}$ for all $x\in X$.

By Theorem \[15101701\] and Lemma \[12151902\].

\[14452602\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ be an invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure where $\Theta$ defined in . Then for all $x\in X$ $$\label{15232602}
\left\{
H\in
\left[
\left(
\prod_{z\in X}^{b}
{\mathfrak{H}}_{z}
\right)_{\diamond}^{x}
\right]_{peq}
\mid
(\forall t\in Y)
(H(\cdot)(t)
\bullet
{\mathcal{E}}(\Theta)
\subseteq
\Gamma^{x}(\pi))
\right\}
\subseteq
\Gamma_{\diamond}^{x}(\xi).$$

Let $v\in{\mathcal{E}}(\Theta)$, $t\in Y$ and $H$ belong to the set in the left side of . Thus by $\exists\,F\in\Gamma(\xi)$ such that $F_{t}
\bullet
v
\in\Gamma(\pi)$, $F(x)=H(x)$ and $H(\cdot)(t)
\bullet
v
\in
\Gamma^{x}(\pi)$ by construction. Then by Corollary \[28111707\] we obtain for all $j\in J$ $$\lim_{z\to x}
\nu_{j}(H(z)(t)v(z)-F(z)(t)v(z)).$$ Therefore the statement follows by Lemma \[15482712\] and .

\[12012802\] Notice that Lemma \[14452602\] holds if we replace invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure with invariant $\left(\Theta,{\mathcal{E}}\right)-$ structure, see Def. \[10282712\], and assume that $Comp(Y)=\{\{t\}\mid t\in Y\}$.

\[21152602\] Assume that ${\mathfrak{V}}$ is a Banach bundle and set ${\mathbf{E}}\doteqdot\{{\mathfrak{E}}_{x}\}_{x\in X}$. By using the notations of Definitions \[16420402\] and \[13361311biss\] assume that $M>1$, $\beta\in{\mathbb {R}}$ and $T\in{\mathcal{G}}(M,\beta,{\mathbf{E}})$ satisfy the property of separation of the spectrum, moreover that there exists a closed curve $\Gamma$ of which in Definition \[13361311biss\] such that $$\label{19281311}
\begin{cases}
Re(\Gamma)
\subseteq
{\mathbb {R}}^{-},
\\
\beta\geq 0
\Rightarrow
-\beta
\notin
Re(\Gamma).
\end{cases}$$ Moreover assume that for all $\mu\in\{\nu,\eta_{s}\mid s\in K(\Gamma)\}$

1.  ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,{\mathbb {R}}^{+}{\right\rangle}}$ is an invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure [^22];

2.  ${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{H}}{\right\rangle}}$ is $\mu-$related;

3.  ${\mathfrak{Q}}
    $ is full;

4.  for all $z\in X$ $$\label{11130307}
    {\mathcal{C}_{cs}
    \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z})\right)}
    \subseteq
    {\mathfrak{H}}_{z};$$

5.  $\Gamma(\xi)
    \bigstar
    {\mathcal{E}}(\Theta)
    \subseteq
    \Gamma(\zeta)$.

Here $\Theta$ is defined in , ${\mathfrak{Q}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{H}},\gamma{\right\rangle}},\xi,X,{\mathfrak{R}}{\right\rangle}}$, ${\mathfrak{Z}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{T}},\gamma{\right\rangle}},\zeta,X,{\mathfrak{K}}{\right\rangle}}$ and ${\mathbf{H}}_{x}\doteqdot{\mathfrak{H}}_{x}$, for all $x\in X$.

Then for all $x\in X$ $$\label{12042802}
{\mathcal{W}}_{T}\in
\Gamma^{x}(\xi)
\Rightarrow
P\bullet
\Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi)
\subseteq
\Gamma^{x}(\pi).$$ Moreover if ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ is an invariant $\left(\Theta,{\mathcal{E}}\right)-$structure such that ${\mathfrak{B}}_{x}={\mathcal{L}}\left({\mathfrak{E}}_{x}\right)$ for all $x$ and ${\mathfrak{D}}$ is full then for all $x\in X$ $$\label{12102802}
{\mathcal{W}}_{T}
\in
\Gamma^{x}(\xi)
\Rightarrow
P
\in\Gamma^{x}(\eta),$$ where ${\mathfrak{D}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{B}},\gamma{\right\rangle}},\eta,X,{\mathfrak{L}}{\right\rangle}}$.

In this proof we denote $R_{T}^{\rho}$ simply by $R^{\rho}$. $R^{\rho}$ is $K(\Gamma)-$supported by construction, moreover by the analyticity of the resolvent map $R^{\rho}(x)$ is $\|\cdot\|_{B({\mathfrak{E}}_{x})}-$ continuous hence continuous as a map valued in ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$. [^23] So $$R^{\rho}
\in\prod_{z\in X}
{\mathcal{C}_{cs}
\left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z})\right)},$$ hence by follows $$\label{17500107}
R^{\rho}
\in
\prod_{z\in X}
{\mathfrak{H}}_{z}.$$ By for all $s\in K(\Gamma)$, $x\in X$ and $\forall v_{x}\in {\mathfrak{E}}_{x}$ $$\label{17051311}
\begin{aligned}
\|
R(T(x),\rho(s))v_{x}
\|
&
\leq
\int_{{\mathbb {R}}^{+}}^{*}
e^{-|Re(\rho(s))|t}
\|e^{-tT(x)}v_{x}\|
dt
\\
&
\leq
M\|v_{x}\|
\int_{{\mathbb {R}}^{+}}^{*}
e^{\left(\beta-|Re(\rho(s))|\right) t}
dt
=
\frac{M\|v_{x}\|}{\beta-|Re(\rho(s))|},
\end{aligned}$$ where $\int_{{\mathbb {R}}^{+}}^{*}$ is the upper integral on ${\mathbb {R}}^{+}$ with respect to the Lebesgue measure. We considered in the first inequality [@IntBourb Prop. $6$, $n^{\circ}$, $\S 1$, Ch. $6$], in the second one the inequality , finally in the equality the Laplace transform of the map $\exp(\beta t)$. Therefore by and $$R^{\rho}
\in\left(
\prod_{z\in X}
{\mathfrak{H}}_{z}
\right)_{peq}.$$ Thus by , and $$\label{10312802}
R^{\rho}
\in\left(
\prod_{z\in X}^{b}
{\mathfrak{H}}_{z}
\right)_{peq}.$$ By and we have that $R^{\rho}\bigstar v
\in
\prod_{z\in X}
{\mathfrak{T}}_{z}$ for all $v\in\prod_{x\in X}^{b}{\mathfrak{E}}_{x}$. By hypothesis we deduce that $\frac{1}
{\beta-|Re(\rho(s))|}$ is defined on $K(\Gamma)$, hence continuous and integrable in it, thus by $$\label{17590207}
R^{\rho}\bigstar v
\in
\left[
\prod_{z\in X}
{\mathfrak{T}}_{z}
\right]_{ui}.$$

By the continuity of $\frac{1}
{\beta-|Re(\rho(s))|}$ on $K(\Gamma)$ we deduce that the map $
\frac{|\frac{d\rho}{ds}(s)|}
{\beta-|Re(\rho(s))|}
$ is integrable in $K(\Gamma)$. Hence by and $$\label{17483006}
\sup_{x\in X}
\|P(x)\|_{B({\mathfrak{E}}_{x})}
\leq
D
\doteqdot
\frac{1}{2\pi i}
\int_{K(\Gamma)}
\frac{M\left|\frac{d\rho}{ds}(s)\right|}
{\beta-|Re(\rho(s))|}
\,ds.$$ Therefore for all $v\in{\mathcal{E}}$ by considering that ${\mathcal{E}}\subset\prod_{x\in X}^{b}{\mathfrak{E}}_{x}$ $$\sup_{x\in X}
\|P(x)v(x)\|_{B({\mathfrak{E}}_{x})}
\leq
D\sup_{x\in X}
\|v(x)\|_{{\mathfrak{E}}_{x}}
<\infty$$ Thus $$\label{15181303}
P\in\prod_{x\in X}^{b}{\mathfrak{B}}_{x}.$$ Let $x\in X$ and $v\in\Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi)$. By for all $s\in K(\Gamma)$ $$\label{17542502}
(R^{\rho}\bigstar v)(x)(s)
=
{\overset{\eta_{s}}{\blacklozenge}}
({\mathcal{W}}_{T}\bigstar v)(x).$$ Moreover by $$\label{17562502}
P\bullet v
=
{\overset{\nu}{\blacklozenge}}
(R^{\rho}\bigstar v).$$ Notice that $(R^{\rho}\bigstar v)(x)(s)
=
\left(R^{\rho}(\cdot)(s)\bullet v
\right)(x)
$ so by for all $s\in K(\Gamma)$ $$\label{18022502}
R^{\rho}(\cdot)(s)\bullet v
=
{\overset{\eta_{s}}{\blacklozenge}}
({\mathcal{W}}_{T}\bigstar v).$$ If ${\mathcal{W}}_{T}\in\Gamma^{x}(\xi)$ then by [@kato Ch. $9$, $\S1$, $n^{\circ}4$, $(1.37)$] and the hyp. that ${\mathfrak{Q}}$ is full follows that ${\mathcal{W}}_{T}
\in
\Gamma_{\diamond}^{x}(\xi)_{peq}$. Therefore by Corollary \[15262502\] and for all $s\in K(\Gamma)$ $$\label{13540303}
R^{\rho}(\cdot)(s)
\bullet v
\in\Gamma^{x}(\pi).$$ By construction ${\mathcal{E}}(\Theta)
\subseteq\Gamma(\pi)$ so ${\mathcal{E}}(\Theta)
\subseteq
\Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi)$, hence $$\label{18042502}
R^{\rho}(\cdot)(s)
\bullet 
{\mathcal{E}}(\Theta)
\subseteq
\Gamma^{x}(\pi).$$ Moreover by the hypothesis that ${\mathfrak{Q}}$ is full and by follows that $$R^{\rho}
\in
\left[
\left(
\prod_{z\in X}^{b}
{\mathfrak{H}}_{z}
\right)_{\diamond}^{x}
\right]_{peq}.$$ Hence by Lemma \[14452602\] and $$\label{18132502}
R^{\rho}
\in\Gamma_{\diamond}^{x}(\xi)_{peq}.$$ Finally follows by , , and Corollary \[15262502\]. In conclusion by , and the hyp. that ${\mathfrak{D}}$ is full $$P\in
\left[
\left(
\prod_{z\in X}^{b}
{\mathfrak{B}}_{z}
\right)_{\diamond}^{x}
\right]_{peq}.$$ Thus follows by , Remark \[12012802\] and by ${\mathcal{E}}(\Theta)
\subseteq
\Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi)$.

\[12500303\] The statements of Corollary \[21152602\] hold by replacing “for all $x\in X$” by “$x_{\infty}\in X$”, and the hypothesis that ${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{H}}{\right\rangle}}$ is $\mu-$related and ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,{\mathbb {R}}^{+}{\right\rangle}}$ is an invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure for all $\mu\in\{\nu,\eta_{s}\mid s\in K(\Gamma)\}$, with the following two

1.  ${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{H}}{\right\rangle}}$ is $\nu-$related and ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,{\mathbb {R}}^{+}{\right\rangle}}$ is an invariant $\left(\Theta,{\mathcal{E}},\nu\right)-$ structure;

2.  $R^{\rho}(\cdot)(s)
    \bullet
    {\mathcal{E}}
    \subseteq
    \Gamma^{x_{\infty}}(\pi)$ for all $s\in K(\Gamma)$.

Indeed follows for $x=x_{\infty}$ and the proof runs as that of Cor. \[21152602\].

The following result shall permit to apply Corollary \[21152602\] to the Main Theorem \[17301812b\].

\[23332802\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure and denote ${\mathfrak{W}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$. Assume that holds for all $x\in X$ and let ${\left\langle {\mathfrak{V}},{\mathfrak{V}}
({\mathbf{M}}^{\mu},\Gamma(\rho)),X,Y{\right\rangle}}$ be the $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure underlying ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$. Then for all $x\in X$ $$\label{17350303}
\Gamma_{\diamond}^{x}(\rho)
\subseteq
\Gamma_{\Gamma(\rho)}^{x}
(\pi_{{\mathbf{M}}^{\mu}})$$ where the right-side of the implication has to be considered modulo the canonical isomorphism.

By Remark \[17150312\] $\Gamma(\rho)
\subseteq
\Gamma(\pi_{{\mathbf{M}}^{\mu}})$, modulo the canonical isomorphism. Thus the statement by Lemma \[15482712\] and .

\[14460503\] Let ${\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\|\cdot\|{\right\rangle}}$ be a Banach bundle. Let $x_{\infty}\in X$ and $
{\mathcal{U}}_{0}
\in\prod_{x\in X_{0}}
{\mathcal{C}_{}
\left({\mathbb {R}}^{+},B_{s}({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x}$ for all $x\in X_{0}$. Moreover let $T_{x}$ be the infinitesimal generator of the semigroup ${\mathcal{U}}_{0}(x)$ for any $x\in X_{0}$. Assume $D(T_{x_{\infty}})$, as defined in Notations \[15411512b\], to be dense in ${\mathfrak{E}}_{x_{\infty}}$, that $\{v(x)\mid v\in{\mathcal{E}}\}$ is dense in ${\mathfrak{E}}_{x}$ for all $x\in X_{0}$ and $\exists\lambda_{0}>0$ (respectively $\exists\lambda_{0}>0,
\lambda_{1}<0$) such that the range ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$, (respectively the ranges ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ and ${\mathcal{R}}(\lambda_{1}-T_{x_{\infty}})$ are dense in ${\mathfrak{E}}_{x_{\infty}}$). Set $$\label{16290303}
\begin{cases}
{\mathcal{T}}_{0}(x)
\doteqdot
Graph(T_{x}),\,
x\in X_{0}
\\
\Phi
\doteqdot
\{
\phi\in\Gamma^{x_{\infty}}
(\pi_{{\mathbf{E}}^{\oplus}})
\mid
(\forall x\in X_{0})
(\phi(x)\in{\mathcal{T}}_{0}(x))
\}
\\
{\mathcal{E}}
\doteqdot
\{
v
\in
\Gamma(\pi)
\mid
(\exists\,\phi\in\Phi)
(v(x_{\infty})=\phi_{1}(x_{\infty}))
\}
\\
\Theta
\doteqdot
\left\{
\prod_{x\in X}\{v(x)\}
\mid
v\in{\mathcal{E}}
\right\}
\\
T_{x_{\infty}}:
D(T_{x_{\infty}})
\ni
\phi_{1}(x_{\infty})
\mapsto
\phi_{2}(x_{\infty}),
\end{cases}$$ where ${\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}^{\oplus}),\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}){\right\rangle}},\pi_{{\mathbf{E}}^{\oplus}},X,{\mathfrak{n}}^{\oplus}{\right\rangle}}$ is the bundle direct sum of the family $\{{\mathfrak{V}},
{\mathfrak{V}}\}$. Set $${\mathcal{U}}\in\prod_{x\in X}
{\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{x})}}
({\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}))$$ (respectively ${\mathcal{U}}\in\prod_{x\in X}
{\mathbf{U}}_{is}
({\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}))$) such that ${\mathcal{U}}(x)$ is the the $C_{0}-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x}$ whose infinitesimal generator is $T_{x}$ for all $x\in X$. The definitions of $T_{x_{\infty}}$ and ${\mathcal{U}}(x_{\infty})$ are well-posed by Theorem \[17301812b\]. Finally let ${\mathcal{T}}$ be defined as in Notations \[15411512b\], with ${\mathcal{T}}_{0}$ and $\Phi$ given in . Recall that by Theorem \[17301812b\] $$\{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}\}
\in
Graph\left({\mathfrak{V}},{\mathfrak{V}}\right).$$

\[**MAIN**${\mathbf{2}}$\] \[13020103\] Let $X$ be compact and ${\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\|\cdot\|{\right\rangle}}$ be a Banach bundle. Let $x_{\infty}\in X$ and $
{\mathcal{U}}_{0}
\in\prod_{x\in X_{0}}
{\mathcal{C}_{}
\left({\mathbb {R}}^{+},B_{s}({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x}$ for all $x\in X_{0}\doteqdot X-\{x_{\infty}\}$. Moreover let $T_{x}$ be the infinitesimal generator of the semigroup ${\mathcal{U}}_{0}(x)$ for any $x\in X_{0}$. Assume that $\{v(x)\mid v\in{\mathcal{E}}\}$ is dense in ${\mathfrak{E}}_{x}$ for all $x\in X$ and $\exists\lambda_{0}>0$ (respectively $\exists\lambda_{0}>0,
\lambda_{1}<0$) such that the range ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$, (respectively the ranges ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ and ${\mathcal{R}}(\lambda_{1}-T_{x_{\infty}})$ are dense in ${\mathfrak{E}}_{x_{\infty}}$). Let us assume the notations of Definition \[14460503\] moreover let ${\mathfrak{Z}}
=
{\left\langle {\left\langle {\mathfrak{T}},\delta{\right\rangle}},\zeta,X,{\mathfrak{K}}{\right\rangle}}$ and ${\mathfrak{W}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ be such that

1.  ${\mathfrak{W}}$ is full and ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ is a $\left(\Theta,{\mathcal{E}}\right)-$structure;

2.  for all $x\in X$ $${\mathcal{C}_{cs}
    \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)}
    \subseteq
    {\mathfrak{M}}_{x}
    \subseteq
    {\mathfrak{L}}_{1}({\mathbb {R}}^{+},
    {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\nu)
    \bigcap
    \bigcap_{\lambda>0}
    {\mathfrak{L}}_{1}({\mathbb {R}}^{+},
    {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\mu_{\lambda});$$

3.  ${\left\langle {\mathfrak{V}},{\mathfrak{V}}({\mathbf{M}}^{\nu},
    \Gamma(\rho)),X,Y{\right\rangle}}$ is invariant and ${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{M}}^{\nu}{\right\rangle}}$ is $\nu-$related;

4.  $\Gamma(\rho)
    \bigstar
    {\mathcal{E}}(\Theta)
    \subseteq
    \Gamma(\zeta)$;

5.  ${\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{x})}}
    ({\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}))
    \subseteq
    {\mathfrak{M}}_{x}$ (respectively ${\mathbf{U}}_{is}
    ({\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}))
    \subseteq
    {\mathfrak{M}}_{x}$), for all $x\in X$;

6.  $\exists\,F\in\Gamma(\rho)$ such that $F(x_{\infty})={\mathcal{U}}(x_{\infty})$ and

    i

    :   ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the ${\mathbf{LD}}_{x_{\infty}}(\{F\},{\mathcal{E}})$;

    ii

    :   $(\forall v\in{\mathcal{E}})
        (\exists\,\phi\in\Phi)$ s.t. $\phi_{1}(x_{\infty})
        =
        v(x_{\infty})$ and $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X
        \mid
        \lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ we have that $\{
        {\mathcal{U}}(z_{n})(\cdot)\phi_{1}(z_{n})
        -
        F(z_{n})(\cdot)v(z_{n})
        \}_{n\in{\mathbb{N}}}$ is a bounded equicontinuous sequence;

    iii

    :   $X$ is metrizable.

Finally assume that $$X\ni x
\mapsto 
-T_{x}\in
Cld({\mathfrak{E}}_{x}),$$ satisfies the property of separation of the spectrum, moreover that there exists a closed curve $\Gamma$ of which in Definition \[13361311biss\], for the position $T(x)=-T_{x}$ for all $x\in X$, and such that $$\begin{cases}
Re(\Gamma)
\subseteq
{\mathbb {R}}^{-},
\\
-1
\notin
Re(\Gamma).
\end{cases}$$

1.  ${\mathcal{P}}
    \bullet
    \Gamma_{{\mathcal{E}}(\Theta)}^{x_{\infty}}(\pi)
    \subseteq
    \Gamma^{x_{\infty}}(\pi)$;

2.  If ${\mathfrak{D}}
    \doteqdot
    {\left\langle {\left\langle {\mathfrak{B}},\tau{\right\rangle}},\eta,X,{\mathfrak{L}}{\right\rangle}}$ is full and ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ is an invariant $\left(\Theta,{\mathcal{E}}\right)-$structure such that $
    \Pr\left({\mathfrak{E}}_{x}\right)
    \subset
    {\mathfrak{B}}_{x}$ for all $x$ then $$\label{19000107}
    {\mathcal{P}}\in\Gamma^{x_{\infty}}(\eta),$$ and $$\label{12102802bis}
    \boxed{
    \{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}\}
    \in
    \Delta
    {\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}},
    }$$ moreover ${\mathcal{P}}$ satisfies .

Here ${\left\langle {\mathfrak{V}},{\mathfrak{V}}({\mathbf{M}}^{\nu},
\Gamma(\rho)),X,{\mathbb {R}}^{+}{\right\rangle}}$ is the $\left(\Theta,{\mathcal{E}},\nu\right)-$ structure underlying ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$. While for all $x\in X$ we set $${\mathcal{P}}(x)
\doteqdot
-
\frac{1}{2\pi i}
\int_{\Gamma}
R(-T_{x};\zeta)\,
d\zeta
\in B({\mathfrak{E}}_{x}),$$ with $R(-T_{x};\cdot):
P(-T_{x})
\ni \zeta
\mapsto
(-T_{x}-\zeta)^{-1}
\in B({\mathfrak{E}}_{x})$ the resolvent map of $-T_{x}$ and $P(-T_{x})$ is its resolvent set, while the integration is with respect to the norm topology on $B({\mathfrak{E}}_{x})$.

By the Dupre’ Thm., see for example [@kurtz Cor. $2.10$], and the fact that a metrizable space is completely regular, we deduce by hyp. $(6.iii)$ that ${\mathfrak{V}}$ is full. So by Prop. \[19492307\] and the density assumption follows that $Dom(T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$. Thus we can use the notations in Def. \[14460503\] and by Theorem \[17301812b\] we have that ${\mathcal{U}}\in
\Gamma^{x_{\infty}}
(\rho)$, where ${\mathcal{U}}(x)$ is the $C_{0}-$semigroup generated by $T_{x}$, for all $x\in X$. Thus by Proposition \[23332802\] and that $F(x_{\infty})={\mathcal{U}}(x_{\infty})$ (by hyp. $(6)$) we have $$\label{20180207}
{\mathcal{U}}
\in
\Gamma_{\diamond}^{x_{\infty}}
(\pi_{{\mathbf{M}}^{\nu}})$$ Moreover ${\mathfrak{W}}$ being full implies ${\mathfrak{M}}_{x}\subset
{\mathbf{M}}_{x}^{\nu}$ so for all $x\in X$ $$\label{20273006}
{\mathcal{C}_{cs}
\left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)}
\subset
{\mathbf{M}}_{x}^{\nu}.$$ Therefore statement $(1)$ follows by Remark \[12500303\] applied to the $\left(\Theta,{\mathcal{E}},\nu\right)-$ structure underlying ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$, by hyp. $(6.i)$ and by the fact that $F\in\Gamma(\pi_{{\mathbf{M}}^{\nu}})$ indeed $\Gamma(\rho)=
\Gamma(\pi_{{\mathbf{M}}^{\nu}})$ modulo the canonical isomorphism, see Rmk. \[17150312\]. Statement $(2)$ follows by , , and ${\mathcal{U}}
=
{\mathcal{W}}_{T}$ for the position $T(x)\doteqdot -T_{x}$ for all $x\in X$.

By follows that is equivalent to $(\exists\, H\in\Gamma(\eta))
(\forall v\in{\mathcal{E}})$ $$\label{18550107}
\lim_{z\to x_{\infty}}
\left\|\left({\mathcal{P}}(z)-H(z)\right)v(z)\right\|
=0.$$

\[17020107\] Let $E$ be a Banach space and $S\subseteq Bounded(E)$ such that $\bigcup_{B\in S}B$ is total in $E$. Set $
\mu_{f}:
{\mathcal{C}_{cs}
\left({\mathbb {R}}^{+}\right)}
\ni
g\mapsto
\int_{{\mathbb {R}}^{+}}f(s)g(s)\,ds$ for all $f\in{\mathfrak{L}}_{\infty}({\mathbb {R}}^{+})$. Then $$\label{19470103}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E))
\subseteq
\bigcap_{f\in{\mathfrak{L}}_{\infty}({\mathbb {R}}^{+})}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathcal{L}}_{S}(E),\mu_{f}).$$ In particular $$\label{19480103}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E))
\subseteq
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathcal{L}}_{S}(E),
\eta_{\lambda}))
\bigcap
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathcal{L}}_{S}(E),\nu)$$

By we deduce that ${\mathfrak{L}}_{\infty}({\mathbb {R}}^{+})
\cdot
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E))
\subseteq
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E))$, hence for all $f\in{\mathcal{L}}_{\infty}({\mathbb {R}}^{+})$. $${\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E))
\subset
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E),\mu_{f}).$$ Moreover by the fact that the norm topology on $B(E)$ is stronger than the lct on ${\mathcal{L}}_{S}(E)$ we have for all $\mu\in Radon({\mathbb {R}}^{+})$ that $${\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E),\mu)
\subseteq
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathcal{L}}_{S}(E),\mu).$$ Hence follows. Finally follows by .

Let ${\mathfrak{V}}
\doteqdot
{\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\|\cdot\|{\right\rangle}}$ be a Banach bundle and $S_{x}\subseteq Bounded({\mathfrak{E}}_{x})$ such that $\bigcup_{B_{x}\in S_{x}}B_{x}$ is total in ${\mathfrak{E}}_{x}$ for all $x\in X$. Set $$\label{20160107a}
{\mathbf{M}}_{x}
\doteqdot
{\mathfrak{L}}_{1}({\mathbb {R}}^{+},B({\mathfrak{E}}_{x}))
\cap
{\mathcal{C}_{}
\left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)}.$$ Thus $$\label{20160107b}
{\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{x})}}
(B({\mathfrak{E}}_{x}))
\subset
{\mathbf{M}}_{x}.$$ Therefore by in Thm. \[13020103\] we can replace hyp. $(2)$ with , while by we can replace “ ${\mathcal{U}}_{0}
\in\prod_{x\in X_{0}}
{\mathcal{C}_{}
\left({\mathbb {R}}^{+},B_{s}({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions” with “ ${\mathcal{U}}_{0}
\in\prod_{x\in X_{0}}
{\mathcal{C}_{}
\left({\mathbb {R}}^{+},B({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $\|\cdot\|-$continuous semigroup of contractions” and delete hyp. $(4)$. Similar replacement can be performed in Thm. \[17301812b\]. Finally notice that in general do not hold if we replace ${\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{x})}}
(B({\mathfrak{E}}_{x}))$ with ${\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{x})}}
({\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}))$, so we cannot take the choice if we want to have results about $C_{0}-$semigroups.

Constructions
=============

 Kurtz Bundle Construction  {#17572301}
---------------------------

In this section we shall construct construct a special bundle ${\mathfrak{E}}$ of Banach space such that for it the Main Theorem \[17301812b\] reduces to the [@kurtz Th. $2.1.$] showing in this way that (a particular case) of the construction of Kurtz falls into the general setting of bundle of $\Omega-$spaces.

In this section we shall use the notations of [@kurtz] with the additional specification of denoting with $\|\cdot\|_{n}$ the norm in the Banach space $L_{n}$. Moreover we denote by $X$ the Alexandrov (one-point) compactification of the locally compact space ${\mathbb{N}}$ with the discrete topology. Here we recall some definitions given in [@kurtz]. ${\left\langle L,\|\cdot\|{\right\rangle}}$ is a Banach space and $\{{\left\langle L_{n},\|\cdot\|_{n}{\right\rangle}}\}_{n\in{\mathbb{N}}}$ is a sequence of Banach spaces, moreover $\{P_{n}\in B(L,L_{n})\}_{n\in{\mathbb{N}}}$ is a sequence of bounded linear operators such that $\forall f\in L$ $$\label{10572601}
\lim_{n\to\infty}
\|P_{n}f\|_{n}=\|f\|.$$ Given an element $f\in L$ and a sequence $\{f_{n}\}_{n\in{\mathbb{N}}}$ such that $f_{n}\in L_{n}$ for all $n\in{\mathbb{N}}$ we set $$\label{11012601}
\lim_{n\to\infty}f_{n}
{\overset{K}{=}}f\overset{def}{\Leftrightarrow}
\lim_{n\to\infty}
\|f_{n}-P_{n}f\|_{n}=0.$$ In addition to the requirements of [@kurtz] we assume also that $$\label{11212601}
(\forall n\in{\mathbb{N}})
({\overline}{P_{n}(L)}=L_{n})$$ We shall set here $L_{\infty}\doteqdot L$, $\|\cdot\|
\doteqdot
\|\cdot\|_{\infty}$, where $\|\cdot\|$ is the norm on $L$. Finally for all $Z$ we recall that $B_{s}(Z)$ is the locally convex space of all linear bounded operators on $Z$ with the strong operator topology.

\[unique\] Let $f,g\in L$ and $\{f_{n}\}_{n\in{\mathbb{N}}}$ such that $f_{n}\in L_{n}$ for all $n\in{\mathbb{N}}$. Then $
(\lim_{n\to\infty}f_{n}
{\overset{K}{=}}f)
\wedge
(\lim_{n\to\infty}f_{n}
{\overset{K}{=}}g)
\Rightarrow
f=g
$

Let $(\lim_{n\to\infty}f_{n}
{\overset{K}{=}}f)$ and $(\lim_{n\to\infty}f_{n}
{\overset{K}{=}}g)$ thus $$\lim_{n\in{\mathbb{N}}}
\|
P_{n}(f-g)
\|
\leq
\lim_{n\in{\mathbb{N}}}
\|P_{n}f-f_{n}\|
+
\lim_{n\in{\mathbb{N}}}
\|P_{n}g-f_{n}\|
=0,$$ so the statement follows by .

\[10452601\] Set $$\begin{cases}
{\mathbf{L}}\doteqdot
\{
{\left\langle L_{x},\|\cdot\|_{x}{\right\rangle}}
\}_{x\in X},
\\
{\mathcal{E}}(L)\doteqdot
\{\sigma^{f}\mid f\in L\},
\end{cases}$$ where $\sigma^{f}\in\prod_{x\in X}L_{x}$ such that $\sigma^{f}(n)\doteqdot P_{n}f$ for all $n\in{\mathbb{N}}$ and $\sigma^{f}(\infty)\doteqdot f$.

\[11322601\] By the sequence $\{\|P_{n}f\|_{n}\}_{n\in{\mathbb{N}}}$ is bounded for all $f\in L$ so $\sigma^{f}\in\prod_{x\in X}^{b}L_{x}$. Moreover by ${\mathcal{E}}(L)$ satisfies $FM(4)$, finally by the request it satisfies also $FM(3)$. Therefore we can define the *Kurtz bundle* the following bundle $${\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L))$$ generated by the couple ${\left\langle {\mathbf{L}},{\mathcal{E}}(L){\right\rangle}}$, see in Def. \[17471910Ba\].

\[16022601\] By Remark \[17150312\] and the compacteness of $X$ by construction, we have that $$\label{16022601b}
{\mathcal{E}}(L)
\simeq
\Gamma(\pi_{{\mathbf{L}}}).$$ Finally by applying the principle of uniform boundedness, [@kato Th. $1.29$, $No 3$, Ch.$3$], we deduce that the sequence $\{\|P_{n}\|_{B(L,L_{n})}\}_{n\in{\mathbb{N}}}$ is bounded.

\[18151602\] Fix $
{\mathcal{U}}_{0}
\in\prod_{n\in{\mathbb{N}}}
{\mathcal{C}_{}
\left({\mathbb {R}}^{+},B_{s}(L_{n})\right)}
$ such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of isometries on $L_{n}$ for all $n\in{\mathbb{N}}$. Denote by $T_{n}$ the infinitesimal generator of the semigroup ${\mathcal{U}}_{0}(n)$ for any $n\in{\mathbb{N}}$. Let us take the positions , where ${\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}^{\oplus}),\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}){\right\rangle}},\pi_{{\mathbf{E}}^{\oplus}},X,{\mathfrak{n}}^{\oplus}{\right\rangle}}$ is the bundle direct sum of the family $\{{\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L)),
{\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L))\}$. In addition we maintain the Notations \[15411512b\] where ${\mathfrak{V}}$ has to be considered the Kurtz bundle and $x_{\infty}\doteqdot\infty$, thus $
{\mathcal{T}}
\in
\prod_{x\in X}
Graph(L_{x}
\times L_{x})
$ so that $
{\mathcal{T}}{\upharpoonright}X-\{\infty\}
\doteqdot
{\mathcal{T}}_{0}
$ and $${\mathcal{T}}(\infty)
\doteqdot
\{\phi(\infty)\mid\phi\in\Phi\},$$ and $
D(T_{\infty})
\doteqdot
\Pr_{1}^{\infty}({\mathcal{T}}(\infty))
=
\{\phi_{1}(\infty)
\mid\phi\in\Phi\}.
$ Finally ${\mathcal{S}}
\doteqdot
\{S_{x}\}_{x\in X}$ where $(\forall B\in\Theta)(\forall x\in X)$ $$\label{11232712bis}
\begin{cases}
{\mathbf{D}}(B,{\mathcal{E}})
\doteqdot
{\mathcal{E}}
\cap
\left(\prod_{x\in X}B_{x}\right)
\\
{\mathcal{B}}_{B}^{x}
\doteqdot
\{v(x)\mid v\in{\mathbf{D}}(B,{\mathcal{E}})\}
\}
\\
S_{x}
\doteqdot
\{{\mathcal{B}}_{B}^{x}\mid B\in\Theta\}.
\end{cases}$$

\[11422601\] Let ${\overline}{f}\in\prod_{x\in X}L_{x}$ Thus $$\lim_{n\to\infty}{\overline}{f}(n){\overset{K}{=}}{\overline}{f}(\infty)
\Leftrightarrow
{\overline}{f}\in\Gamma^{\infty}(\pi_{{\mathbf{L}}}).$$

By and implication $(3)\Rightarrow(1)$ of Corollary \[28111707\] we have that $\lim_{n\to\infty}{\overline}{f}(n){\overset{K}{=}}{\overline}{f}(\infty)$ implies that $${\overline}{f}
\text{
is continuous
at $\infty$,}$$ indeed $\sigma^{{\overline}{f}(\infty)}\in
\Gamma(\pi_{{\mathbf{L}}})$ modulo isomorphism. By the upper semicontinuity of $\|\cdot\|:{\mathfrak{E}}\to{\mathbb {R}}^{+}$, due to the construction of the bundle ${\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L))$ and to [@kurtz $1.6.(ii)$], and by the fact that the composition of any u.s.c. map with any continuous one at a point is an u.s.c. map in the same point, we deduce that $\|\cdot\|\circ{\overline}{f}$ is u.s.c. at $\infty$. Thus $\sup_{x\in X}\|{\overline}{f}(x)\|_{x}<\infty$, indeed we applied to the u.s.c. map $\|\cdot\|\circ{\overline}{f}$ the fact that $X$ is compact (so quasi compact), $-\|\cdot\|\circ{\overline}{f}$ is l.s.c, the [@BourGT Th. $3$, $\S 6.2.$, Ch. $4$] and [@BourGT form.$(2)$, $\S 5.4.$, Ch. $4$]. Therefore $${\overline}{f}\in\prod_{x\in X}^{b}L_{x}.$$ Then ${\overline}{f}\in\Gamma^{\infty}(\pi_{{\mathbf{L}}})$. The remaining implication follows by Corollary \[28111707\] and by the fact that ${\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L))$ is full, $X$ being locally compact and by the fact that any Banach bundle over a locally compact space is full, see [@fell Appendix C].

\[15512601\] We have $$\Gamma^{\infty}(\pi_{\mathbf{L}^{\oplus}})
=
\left\{
\sigma_{1}\oplus\sigma_{2}
\mid
\sigma_{i}\in\prod_{x\in X}
L_{x},
\lim_{n\to\infty}\sigma_{i}(n)
{\overset{K}{=}}\sigma(\infty),
i=1,2
\right\}.$$ Here, we used the Convention \[16392601\] and set $(\sigma_{1}\oplus\sigma_{2})(x)
\doteqdot
\sigma_{1}(x)\oplus\sigma_{2}(x)$.

By Convention \[16392601\] and Corollary \[17571212\] $\sigma_{1}\oplus\sigma_{2}$ is continuous at $\infty$ if and only if $\sigma_{i}$ is continuous at $\infty$ for all $i=1,2$. Thus the statement by Proposition \[11422601\].

\[18542601\] Let $
{\mathcal{U}}_{0}
\in\prod_{n\in{\mathbb{N}}}
{\mathcal{C}_{}
\left({\mathbb {R}}^{+},B_{s}(L_{n})\right)}
$ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions on $L_{n}$ for all $n\in{\mathbb{N}}$. Moreover let us denote by $T_{n}$ the infinitesimal generator of the semigroup ${\mathcal{U}}_{0}(n)$ for any $n\in{\mathbb{N}}$. Thus with the positions where ${\mathfrak{V}}$ is the Kurtz bundle we have $$\label{18452801}
\begin{cases}
\Phi
=
\left\{
\sigma_{1}\oplus\sigma_{2}
\mid
(\forall i\in\{1,2\})
(\sigma_{i}\in\prod_{x\in X}
L_{x})(1-2)
\right\}
\\
(1)
\lim_{n\to\infty}\sigma_{i}(n)
{\overset{K}{=}}\sigma_{i}(\infty)
\\
(2)
(\forall n\in{\mathbb{N}})
(\sigma_{1}(n),\sigma_{2}(n))
\in
Graph(T_{n}),
\end{cases}$$ and $$\label{17472801}
\begin{cases}
{\mathcal{E}}
=
\left\{
\sigma^{\sigma_{1}(\infty)}
\mid
\sigma_{1}\in\prod_{x\in X}
L_{x}
(1-2-3)
\right\}
\\
(1)
\lim_{n\to\infty}\sigma_{1}(n)
{\overset{K}{=}}\sigma_{1}(\infty)
\\
(2)
(\forall n\in{\mathbb{N}})
(\sigma_{1}(n)
\in
Dom(T_{n}))
\\
(3)
(\exists\,f\in L_{\infty})
(\lim_{n\to\infty}T_{n}
\sigma_{1}(n)
{\overset{K}{=}}f).
\end{cases}$$ Moreover $\exists\,!\,f$ satisfying $(3)$ in and $(\forall\sigma_{1}\in{\mathcal{E}})
((\sigma_{1},\sigma_{2})\in\Phi)$, where $\sigma_{2}\in\prod_{x\in X}L_{x}$ such that $(\forall n\in{\mathbb{N}})
(\sigma_{2}(n)\doteqdot T_{n}\sigma_{1}(n))$ and $\sigma_{2}(\infty)
\doteqdot f$.

The first sentence follows by Proposition \[15512601\], while the second comes by the first one and Lemma \[unique\].

We assume $\exists\,
\{I_{n}\in B(L_{n},L)\}_{n\in{\mathbb{N}}}$ such that $$\label{15052601}
\begin{cases}
\sup_{n\in{\mathbb{N}}}\|I_{n}\|_{B(L_{n},L)}
<\infty,
\\
(\forall f\in L)
(\forall n\in{\mathbb{N}})
(I_{n}\circ P_{n}=Id).
\end{cases}$$ Moreover we assume that $$\label{11442701}
\varlimsup_{n\to\infty}
\|P_{n}\|\leq 1.$$ In addition we assume that $
(\forall g\in L)
(\exists\,
\sigma_{1}\in\prod_{x\in X}
L_{x})
$ such that $$\label{17202701}
\begin{cases}
(1)
\lim_{n\to\infty}
\sigma_{1}(n)
{\overset{K}{=}}\sigma_{1}(\infty)
\\
(2)
(\forall n\in{\mathbb{N}})
(\sigma_{1}(n)
\in
Dom(T_{n}))
\\
(3)
(\exists\,f\in L_{\infty})
(\lim_{n\to\infty}T_{n}
\sigma_{1}(n)
{\overset{K}{=}}f)
\\
(4)
g=\sigma_{1}(\infty).
\end{cases}$$ Set $$\label{16582801}
{\mathfrak{U}}
\doteqdot
\left\{
F\in
{\mathcal{C}_{}
\left({\mathbb {R}}^{+},B_{s}(L)\right)}
\mid
(\forall s\in{\mathbb {R}}^{+})
(\forall v\in L)
(\|F(s)v\|=\|v\|)
\right\}.$$

In the following definition we shall give the data for constructing a bundle ${\mathfrak{W}}$ such that ${\left\langle {\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L)),
{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ would be a $\left(\Theta,{\mathcal{E}}\right)-$structure.

\[13352701\] Set $P_{\infty}\doteqdot 
I_{\infty}
\doteqdot
Id:L\to L$, moreover $\forall U\in{\mathfrak{U}}$ set $F_{U}\in\prod_{x\in X}
{\mathcal{C}_{c}
\left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}(L_{x})\right)}$ such that $\forall x\in X$ $$\begin{cases}
F_{U}(x)
\doteqdot
P_{x}\circ 
U(\cdot)
\circ I_{x},
\\
P_{x}\circ 
U(\cdot)
\circ I_{x}:
{\mathbb {R}}^{+}\ni s
\mapsto
P_{x}\circ 
U(s)
\circ I_{x}
\in B(L_{x}).
\end{cases}$$ Now we can define $\forall x\in X$ $${\mathbf{M}}_{x}
\doteqdot
\textrm{span}
\left
\{
F_{U}(x)
\mid
U\in{\mathfrak{U}}
\right
\}.$$ ${\mathbf{M}}_{x}$ has to be considered as Hlcs with the topology induced by that on ${\mathcal{C}_{c}
\left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}(L_{x})\right)}$. [^24] Moreover set $${\mathcal{M}}
\doteqdot
\textrm{span}
\left
\{
F_{U}
\mid
U\in{\mathfrak{U}}
\right
\}.$$

\[14282701\] ${\mathbf{M}}_{x}$ as Hlcs is well-defined for any $x\in X$, moreover ${\mathcal{M}}\subset
\prod_{x\in X}^{b}
{\mathbf{M}}_{x}$ and ${\mathbf{M}}_{x}=
\{F(x)\mid F\in{\mathcal{M}}\}$. Finally ${\mathcal{M}}$ satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{M}}$.

By Remark \[21500412b\] we have that ${\mathcal{C}_{c}
\left({\mathbb {R}}^{+},B_{s}(L_{x})\right)}
\subset
{\mathcal{C}_{c}
\left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}(L_{x})\right)}$ hence for the first sentence of the statement it is sufficient to show that $
P_{x}
\circ 
U(\cdot)
\circ I_{x}
\in
{\mathcal{C}_{c}
\left({\mathbb {R}}^{+},B_{s}(L_{x})\right)}$ for any $U\in{\mathfrak{U}}$. For $x=\infty$ is trivial so let $n\in{\mathbb{N}}$ and $f_{n}\in L_{n}$ thus for all $s\in{\mathbb {R}}^{+}$ and all net $\{s_{\alpha}\}_{\alpha\in D}$ in ${\mathbb {R}}^{+}$ converging at $s$ we have $$\lim_{\alpha\in D}
\|
P_{n}
\circ 
U(s_{\alpha})
\circ 
I_{n}
(f_{n})
-
P_{n}
\circ 
U(s)
\circ 
I_{n}
(f_{n})
\|_{n}
=
\lim_{\alpha\in D}
\|
P_{n}
(U(s_{\alpha})-U(s))
I_{n}f_{n}
\|_{n}
=0,$$ where we used the fact that $U$ is strongly continuous and $P_{n}$ is norm continuous by construction. Thus the first sentence of the statement follows. Let $v\in{\mathcal{E}}$ and $U\in{\mathfrak{U}}$ thus $\forall K\in Compact({\mathbb {R}}^{+})$ $$\begin{aligned}
{1}
\sup_{n\in{\mathbb{N}}}
\sup_{s\in K}
\|
P_{n}
U(s)
I_{n}
v(n)
\|_{n}
&\leq
M
\sup_{n\in{\mathbb{N}}}
\sup_{s\in K}
\|
U(s)
I_{n}
v(n)
\|_{\infty}
\\
&=
M
\sup_{n\in{\mathbb{N}}}
\|
I_{n}
v(n)
\|_{\infty}
\\
&\leq
M
\sup_{n\in{\mathbb{N}}}
\|
I_{n}
\|
\sup_{n\in{\mathbb{N}}}
\|
v(n)
\|_{\infty}<\infty.\end{aligned}$$ Here $M\doteqdot\sup_{n\in{\mathbb{N}}}\|P_{n}\|$, in the second one the hypothesis that $U(s)$ is an isometry for all $s\in{\mathbb {R}}^{+}$, in the final inequality we considered , ${\mathcal{E}}\subset\prod_{x\in X}^{b}L_{x}$ and that $M<\infty$ by Remark \[16022601\]. Therefore by Remark \[21500412b\] ${\mathcal{M}}\subset
\prod_{x\in X}^{b}
{\mathbf{M}}_{x}$. The equality ${\mathbf{M}}_{x}=
\{F(x)\mid F\in{\mathcal{M}}\}$ comes by construction, in particular ${\mathcal{M}}$ satisfies the $FM(3)$ with respect to the ${\mathbf{M}}$. $\forall v\in{\mathcal{E}}$ $$\begin{aligned}
{2}
\varlimsup_{n\to\infty}
\sup_{s\in K}
\|
P_{n}U(s)I_{n}
v(n)
\|_{n}
&\leq
\varlimsup_{n\to\infty}
\left(\|
P_{n}
\|
\sup_{s\in K}
\|
U(s)I_{n}
v(n)
\|_{n}
\right),
&
\text{\cite[Prop. $11$, $\S 5.6.$
Ch. $4$]{BourGT}}
\\
&\leq
\varlimsup_{n\to\infty}
\|P_{n}\|
\varlimsup_{n\to\infty}
\sup_{s\in K}
\|U(s)I_{n}v(n)\|_{n},
&
\text{\cite[Prop. $13$, $\S 5.7.$
Ch. $4$]{BourGT}}
\\
&\leq
\varlimsup_{n\to\infty}
\|I_{n}v(n)\|_{\infty},
&
\text{
\eqref{11442701},
\eqref{16582801}}
\\
&=
\varlimsup_{n\to\infty}
\|
I_{n}P_{n}f
\|_{\infty},
&
\text{
$v\in{\mathcal{E}}\subset\Gamma(\pi)
\simeq{\mathcal{E}}(L)$
}
\\
&=
\|f\|_{\infty},
&
\text{
\eqref{15052601}
}
\\
&=
\|v(\infty)\|_{\infty}.\end{aligned}$$ Thus by considering that $U$ is a map of isometries we have $$\varlimsup_{n\to\infty}
\sup_{s\in K}
\|
P_{n}U(s)I_{n}
v(n)
\|_{n}
\leq
\sup_{s\in K}
\|
P_{\infty}
U(s)I_{\infty}
v(\infty)
\|_{\infty}.$$ Hence by [@BourGT Prop. $3$, $\S 7.1.$ Ch. $4$] and [@BourGT $(13)$, $\S 5.6.$ Ch. $4$] we deduce that $$X\ni x\mapsto
\sup_{s\in K}
\|
P_{x}U(s)I_{x}
v(x)
\|_{x}
\text{ is $u.s.c.$ at $\infty$},$$ therefore it is $u.s.c.$ on $X$ because of it is continuous in each point in ${\mathbb{N}}$ due to the fact that the topology induced on ${\mathbb{N}}$ by that on $X$ is the discrete topology. So ${\mathcal{M}}$ satisfies the $FM(4)$ with respect to the ${\mathbf{M}}$.

\[16232701\] Theorem \[14282701\] allows us to construct a bundle of $\Omega-$space namely the bundle ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ generated by the couple ${\left\langle {\mathbf{M}},{\mathcal{M}}{\right\rangle}}$, see Def \[17471910Ba\].

\[16282701\] By Remark \[17150312\] and the compactness of $X$ we have $$\label{14222801}
{\mathcal{M}}
\simeq
\Gamma(\pi_{{\mathbf{M}}}).$$ Hence by ${\mathbf{M}}_{x}=
\{F(x)\mid F\in{\mathcal{M}}\}$ we have that ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ is full.

\[16332701\] We have that ${\overline}{\bigcup_{B\in\Theta}{\mathcal{B}}_{B}^{x}}
=L_{x}$ for all $x\in X$ moreover ${\left\langle {\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L)),
{\mathfrak{V}}({\mathbf{M}},{\mathcal{M}}),X,{\mathbb {R}}^{+}{\right\rangle}}$ is a $\left(\Theta,{\mathcal{E}}\right)-$structure.

By assumptions , , Proposition \[18542601\] and Remark \[21500412b\] we obtain that ${\overline}{\bigcup_{B\in\Theta}{\mathcal{B}}_{B}^{x}}
=
L_{x}$ for all $x\in X$. The remaining requests for the second sentence of the statement come by the construction of ${\mathcal{M}}$ and ${\mathbf{M}}$.

\[13560202\] If $D(T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$, and $\exists\lambda_{0}>0,
\lambda_{1}<0$ such that the ranges ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ and ${\mathcal{R}}(\lambda_{1}-T_{x_{\infty}})$ are dense in ${\mathfrak{E}}_{x_{\infty}}$), then ${\left\langle {\mathcal{T}},\infty,\Phi{\right\rangle}}
\in
Graph\left(
{\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L)),
{\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L))
\right)$ and the following $$T_{\infty}:D(T_{\infty})
\ni\phi_{1}(\infty)
\mapsto
\phi_{2}(\infty)$$ is a well-defined operator which is the generator of a $C_{0}-$semigroup of isometries on ${\mathfrak{E}}_{\infty}$.

By Propositions \[16332701\] and \[19272701\] we have that the first part of hypotheses of Theorem \[17301812b\] is satisfied so the statement by the first sentence of the statement of Theorem \[17301812b\].

\[16002801\] Let us denote by ${\mathcal{U}}_{\infty}$ the $C_{0}-$semigroup of isometries on $L_{\infty}$. Moreover set ${\mathcal{U}}\in
\prod_{x\in X}{\mathbf{U}}_{is}(B_{s}(L_{x}))$ such that ${\mathcal{U}}{\upharpoonright}{\mathbb{N}}={\mathcal{U}}_{0}$ and ${\mathcal{U}}(\infty)
=
{\mathcal{U}}_{\infty}$.

\[17332801\] $(\exists\,F\in\Gamma(\pi_{{\mathbf{M}}}))
(F(\infty)={\mathcal{U}}(\infty))$ such that $(\forall v\in{\mathcal{E}})
(\exists\,\phi\in\Phi)$ s.t. $\phi_{1}(x_{\infty})
=
v(x_{\infty})$ and $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X
\mid
\lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ we have that $\{
{\mathcal{U}}(z_{n})(\cdot)\phi_{1}(z_{n})
-
F(z_{n})(\cdot)v(z_{n})
\}_{n\in{\mathbb{N}}}$ is a bounded equicontinuous sequence. Moreover we can choose $F$ such that $F=
F_{{\mathcal{U}}_{\infty}}$.

By Prop. \[18542601\] and the statement is equivalent to show that $
\forall
\sigma_{1}\in\prod_{x\in X}
L_{x}$ satisfying $(1-2-3)$ of and $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X
\mid
\lim_{n\in{\mathbb{N}}}z_{n}=\infty)$ we have that $$\label{17592801}
\{
{\mathcal{U}}(z_{n})(\cdot)\sigma_{1}(z_{n})
-
F_{{\mathcal{U}}_{\infty}}
(z_{n})(\cdot)
\sigma^{\sigma_{1}(\infty)}(z_{n})
\}_{n\in{\mathbb{N}}}$$ is a bounded equicontinuous sequence. Moreover by the second assumption and $$\label{18022801}
\{
{\mathcal{U}}(z_{n})(\cdot)
\sigma_{1}(z_{n})
-
P_{z_{n}}
{\mathcal{U}}_{\infty}
(z_{n})(\cdot)
\sigma_{1}(\infty)
\}_{n\in{\mathbb{N}}}$$ is a bounded equicontinuous sequence. Set $\sigma_{2}\in\prod_{x\in X}L_{x}$ such that $\sigma_{2}(x)\doteqdot T_{x}\sigma_{1}(x)$, for all $x\in X$, thus $$\sigma_{i}\in\Gamma^{\infty}(\pi_{{\mathbf{L}}}),$$ for all $i=1,2$, indeed for $i=1$ follows by $(1)$ of and Prop. \[11422601\], while for $i=2$ follows by construction of of $T_{\infty}$, the second sentence of Prop. \[18542601\], the fact that by construction $\Phi\subseteq\Gamma(\pi_{{\mathbf{E}}^{\oplus}})$, see , and finally by Corollary \[17571212\]. Therefore in particular $\sigma_{i}$ is continuous at $\infty$. Thus by considering that $\sigma^{\sigma_{i}(\infty)}
\in\Gamma(\pi_{{\mathbf{L}}})$ modulo isomorphism by , that ${\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L))$ is full being a Banach bundle over a locally compact space, we deduce by Prop. \[28111555\] $$\lim_{n\in{\mathbb{N}}}
\|
\sigma_{i}(z_{n})
-
\sigma^{\sigma_{i}(\infty)}
(\pi\circ\sigma_{i}(z_{n}))
\|_{\pi\circ\sigma_{i}(z_{n})}
=0.$$ Then by considering that $\pi\circ\sigma_{i}=Id$ because of $\sigma_{i}$ is a selection, we have $$\label{19172801}
\lim_{n\in{\mathbb{N}}}
\|
\sigma_{i}(z_{n})
-
P_{z_{n}}\sigma_{i}(\infty)
\|_{z_{n}}
=0.$$ The statement now follows by , and by using the same argumentation used in proof of [@kurtz Th. $1.2$] for proving a similar result.

\[19272701\] With the notations of Def. \[15062301\] we have that $${\mathbf{M}}_{x}
\subset
\bigcap_{\lambda>0}
{\mathfrak{L}}_{1}
({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}(L_{x});
\mu_{\lambda}),$$ and holds.

By Proposition \[18390901\].

\[13512801\] $
{\left\langle {\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L)),
{\mathfrak{V}}({\mathbf{M}},{\mathcal{M}}),X,{\mathbb {R}}^{+}{\right\rangle}}$ has the full Laplace duality property, moreover $\forall U\in{\mathbf{U}}_{1,\|\cdot\|}(B_{s}(L))$, $\forall\lambda>0$ and $\forall f\in L$ we have that $${\mathfrak{L}}(F_{U})(\cdot)(\lambda)
\sigma^{f}(\cdot)
=
\sigma^{(\lambda-T_{U})^{-1}f}.$$ Here $T_{U}$ is the infinitesimal generator of the semigroup $U$.

Let $f\in L$ and $U\in{\mathfrak{U}}$ thus for all $x\in X$ and $\lambda>0$ we have $$\begin{aligned}
{2}
\label{14242801}
\int_{0}^{\infty}
e^{-\lambda s}
P_{x}U(s)I_{x}\sigma^{f}(x)\,
ds
&=
\int_{0}^{\infty}
e^{-\lambda s}
P_{x}U(s)f\,ds
\notag
\\
&=
P_{x}
\int_{0}^{\infty}
e^{-\lambda s}
U(s)f\,ds,\end{aligned}$$ where the first equality follows by the second assumption \[15052601\], while the second one by the linearity and continuity of $P_{x}$ and by . Thus the first sentence of the statement by and . The second sentence of the statement folllows by the and by Hille-Yosida Theorem, see [@kurtz Th. $1.2.$].

\[12560202\] Let us assume the hypotheses of Corollary \[13560202\]. Then $(\forall g\in L)
(\forall K\in Compact({\mathbb {R}}^{+}))$ $$\label{16180202}
\lim_{z\to\infty}
\sup_{s\in K}
\left\|
\left({\mathcal{U}}(z)(s)\circ P_{z}
-
P_{z}\circ{\mathcal{U}}_{\infty}(s)\right)
g
\right\|=0.$$ Moreover $$\label{02512912pbis}
{\mathcal{U}}
\in\Gamma^{\infty}(\rho).$$ In particular $$\label{02512912bis}
\{{\left\langle {\mathcal{T}},\infty,\Phi{\right\rangle}}\}
\in
\Delta_{\Theta}{\left\langle {\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}),
{\mathfrak{V}}({\mathbf{M}},{\mathcal{M}}),{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}.$$

By Proposition \[19272701\] follows , hypothesis $(i)$ of Theorem \[17301812b\] follows by Theorem \[13512801\], $(ii)$ by Theorem , finally $(iii)$ follows by and by the fact that $\{\{n\}\mid n\in{\mathbb{N}}\}$ is a base for the topology on ${\mathbb{N}}$. Thus by Theorem \[17301812b\] we obtain , and $(\forall v\in{\mathcal{E}})
(\forall K\in Compact({\mathbb {R}}^{+}))$ $$\label{16170202}
\lim_{z\to\infty}
\sup_{s\in K}
\left\|
{\mathcal{U}}(z)(s)v(z)
-
F(z)(s)v(z)
\right\|=0,$$ where $F$ is any map of which in Theorem \[17332801\]. Now by Theorem \[17332801\] we can take in the previous equation $F=F_{{\mathcal{U}}_{\infty}}$, moreover by and assumption we have $${\mathcal{E}}=\{\sigma^{g}\mid g\in L\},$$ therefore by $\forall s\in{\mathbb {R}}^{+}$, $\forall z\in X$ and $\forall g\in L$ $$F_{{\mathcal{U}}_{\infty}}(z)
\sigma^{g}(z)
=
(P_{z}\circ{\mathcal{U}}_{\infty}(s))
g.$$ Hence by follows .

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[^1]: Later we shall see that the topology on the bundle space of ${\mathfrak{V}}\oplus{\mathfrak{V}}$ will be constructed in order to ensure that the limit in is equivalent to say that $v=
    \lim_{x\to x_{\infty}}
    \phi_{1}(x)$ and $T_{x_{\infty}}v=
    \lim_{x\to x_{\infty}}
    \phi_{2}(x)$, both limits with respect to the topology on the bundle space ${\mathfrak{E}}$.

[^2]: Indeed it is sufficient to take $Y=\{pt\}$ i.e. one point.

[^3]: Notice the strong similarity of with .

[^4]: Indeed if we set assume that there exists for every $n\in{\mathbb{N}}$ $S_{n}\in B(L_{n},L)$ such that $S_{n} Q_{n}= Id$ then would become $$\label{17520603}
    (\forall t\in{\mathbb {R}}^{+})
    (\forall f\in L)
    (\lim_{n\to\infty}\sup_{0\leq s\leq t}
    \|
    (U_{n}(s)
    -
    Q_{n}U(s)S_{n})
    Q_{n}f
    \|_{n}=0).$$ Moreover let ${\left\langle {\mathfrak{M}},\rho,X{\right\rangle}}$ and ${\left\langle {\mathfrak{E}},\pi,X{\right\rangle}}$ be set as in the beginning and assume that $\{\nu_{(K,v)}^{z}
    \mid(K,v)\in Comp(Y),v\in{\mathcal{E}}
    \}$ is a fundamental set of seminorms on ${\mathfrak{M}}_{z}$ for every $z\in X$, where ${\mathcal{E}}\subseteq\Gamma(\pi)$. Finally assume that for all $K\in Comp(Y)$, $v\in{\mathcal{E}}$ and for all $z\in X$ and $f^{z}\in{\mathfrak{M}}_{z}$ $$\label{21310603}
    \nu_{(K,v)}^{z}(f^{z})
    \doteqdot
    \sup_{s\in K}
    \|
    f^{z}(s)
    v(z)
    \|_{z}.$$ Thus would read: if there exists $\sigma\in\Gamma(\rho)$ such that $\sigma(x_{\infty})=F(x_{\infty})$ then $$\label{17510603}
    F\in\Gamma^{x_{\infty}}(\rho)
    \Leftrightarrow
    (\forall K\in Comp(Y))
    (\forall v\in{\mathcal{E}})
    (\lim_{z\to x_{\infty}}
    \sup_{s\in K}
    \|(F(z)-\sigma(z))v(z)\|_{z}
    =0).$$ Therefore by setting $X$ the Alexandroff compactification of ${\mathbb{N}}$, $x_{\infty}=\infty$ and for all $n\in{\mathbb{N}}$ $$\label{22270603}
    \begin{cases}
    {\mathfrak{E}}_{n}
    \doteqdot
    L_{n},\,
    {\mathfrak{E}}_{\infty}\doteqdot L
    \\
    {\mathfrak{M}}_{n}\doteqdot
    {\mathcal{C}_{c}
    \left({\mathbb {R}}^{+},B_{s}(L_{n})\right)}
    \\
    {\mathfrak{M}}_{\infty}\doteqdot
    {\mathcal{C}_{c}
    \left({\mathbb {R}}^{+},B_{s}(L)\right)}
    \\
    {\mathcal{E}}
    \doteqdot
    \left\{
    Qf
    \mid f\in L
    \right\},
    \end{cases}$$ if there exist conditions under which we can obtain that $$\label{21420603}
    \begin{cases}
    \left\{
    Qf
    \mid f\in L
    \right\}
    \subseteq
    \Gamma(\pi)
    \\
    \left\{
    QVS
    \mid
    V\in{\mathbf{U}}(L)
    \right\}
    \subseteq
    \Gamma(\rho),
    \end{cases}$$ where $
    (Qf)(n)
    \doteqdot
    Q_{n}f
    $, $(Qf)(\infty)
    \doteqdot
    f$, while $(QVS)(n)
    \doteqdot
    Q_{n}V S_{n}$, $(QVS)(\infty)
    \doteqdot
    V$, for all $n\in{\mathbb{N}}$ and ${\mathbf{U}}(L)$, is the class of all $C_{0}-$semigroup on $L$, then by and follows that $${\mathcal{U}}
    \in
    \Gamma^{\infty}(\rho),$$ where ${\mathcal{U}}(n)\doteqdot U_{n}$ and ${\mathcal{U}}(\infty)\doteqdot U$.

[^5]: Standard in the following sense $
    (B,v)
    \mapsto
    B(v)$.

[^6]: Notice the similarity of notation with Def. \[15012602\]. In any case it will be always clear which definition has to be considered.

[^7]: I.e. $
    (\forall j_{1},j_{2}\in J)
    (\exists\,j\in J)
    (\mu_{j_{1}}^{x},\mu_{j_{1}}^{x}
    \leq
    \mu_{j}^{x})
    $ with the standard order relation of pointwise order on $
    {\mathbb {R}}^{V_{x}}
    $.

[^8]: By applying $5.3.$ of [@gie] and Ch.1. of [@BourGT] we know that such as topology there exists.

[^9]: I.e. $\sigma\leftrightarrow f$ iff $
    \sigma(x)
    =(x,
    f(x))
    $

[^10]: An example is when ${\mathfrak{V}}$ is the trivial bundle.

[^11]: See Remark \[22550110\].

[^12]: which ensures that the locally convex topology on $
    \left({\mathfrak{E}}_{i}\right)_{x}
    $ generated by the set of seminorms ${\mathfrak{N}}_{i}^{x}$ is exactly the topology induced on it by the topology $\tau_{i}$ on ${\mathfrak{E}}_{i}$, for all $i$ and $x\in X$.

[^13]: See [@BourGT Def $1$, $\S 2.1$, Ch. $10$] for the definition of equicontinuous sets.

[^14]: Well set indeed by Prop. \[19492307\], the density assumptions and Rem. \[21500412b\] we have that $S_{x}$ is dense in ${\mathfrak{E}}_{x}$ for all $x\in X$.

[^15]: See Proposition \[18390901\] for models of ${\mathfrak{M}}$ satisfying and ${\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{z})}}
    ({\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z}))
    \subseteq
    {\mathfrak{M}}_{z}$.

[^16]: In others words ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}
    =
    {\mathcal{L}}_{{\mathcal{S}}_{x}}({\mathfrak{E}}_{x})$, see Notations \[notat\] and Def. \[17471910A\].

[^17]: Equivalently $-]\beta,\infty[
    \subseteq 
    P(T(x))$, where $P(T(x))$ is the resolvent set of any closed operator $T_{x}$.

[^18]: By identifying ${\mathbb {C}}$ with ${\mathbb {R}}^{2}$, so $\rho$ is derivable with contiuous derivative

[^19]: See Def. \[15531102\] for $(\bullet)$.

[^20]: which is again an algebra by Lemma \[16141302\]

[^21]: In case ${\mathfrak{E}}_{x}$ is a Banach space $Meas(Y,{\mathfrak{E}}_{x},\mu)
    \bigcap
    {\mathfrak{L}}_{1}(Y,{\mathfrak{E}}_{x},\mu)_{s}
    =
    {\mathfrak{L}}_{1}(Y,{\mathfrak{E}}_{x},\mu)_{s}$.

[^22]: See Prop. \[18390901\] or more in general [@SilInt Corollary $2.6.$].

[^23]: Indeed for all $E$ normed space the topology on $B(E)$ is stronger of that on ${\mathcal{L}}_{S}(E)$. For that it is sufficient to show $B(E)={\mathcal{L}}_{Bounded(E)}(E)$. To this end note that $\sup_{v\in E_{r}}
    \|Av\|
    \leq
    r\sup_{v\in E_{1}}
    \|Av\|$ for all $r>0$ and $A\in{\mathcal{L}}(E)$, where $E_{r}=\{v\in E\mid\|v\|\leq r\}$.

[^24]: ${\mathcal{C}_{c}
    \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}(L_{x})\right)}$ is Hausdorff for all $x\in X$ by the fact that $\bigcup_{B\in\Theta}{\mathcal{B}}_{B}^{x}=L_{x}$, see later Prop. \[16332701\].
