Take the 2-minute tour ×  If $\mathcal{A}$ is a complete locally
convex (Hausdorff) associative unital algebra (over
$\mathbb{C}$) one is interested in defining "transcendental"
functions of a given algebra element $a \in \mathcal{A}$ like
e.g. exponentials $\exp(a)$ by means of the power series
expansion. This works fine for complete locally multiplicatively
convex algebras. Recall that $\mathcal{A}$ is called lmc if
there is a defining system of seminorms which are
submultiplicative for the product. Equivalently, such an algebra
is a (suitable) projective limit of Banach algebras. Then the
(algebraic) polynomial calculus sending a polynomial $p \in
\mathbb{C}[z]$ to the algebra element $p(a)$ extends by
completion to an entire calculus $$ \mathcal{O}(\mathbb{C}) \ni
f \mapsto f(a) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} a^n \in
\mathcal{A}, $$ which is a continuous algebra homomorphism for a
given $a$. Here $\mathcal{O}(\mathbb{C})$ is equipped with its
usual Fréchet topology of locally uniform convergenc