Take the tour ×   1. Let V be a Nuclear space (not necessary
Frechet, but complete). Let V^* be the dual space considered
with the weak topology (i.e. pointwise convergence). Is it true
that $V^*$ is also nuclear?    2. Is it true that the map to the
double dual $V \to V^{**}$ is a linear isomorphism (not
necessary continuously invertible)?    3. Let $W$ be a space
like $V$. Is the map $V^* \hat \otimes W \to Hom(V,W)$ is a
linear isomorphism? Here $V^* \hat{ \otimes} W $ is the
completed tensor product and $Hom(V,W)$ is the space of
continuous linear maps. If yes what topology does it induce on
$Hom(V,W)$? If not what are the kernel and the image of this
map?    4. The same questions for different kinds of duals.
share|improve this question Many of your questions are answered
in Section 10a of the book by Kevin Costello and Owen Gwilliam
“Factorization algebras in perturbative quantum field theory”
(math.northwestern.edu/~costello/factorization_public.html) – 
Dmitri Pavlov Nov 22 '11 at 20:20 add comment