30.7 Power Series

Definition 30.7.1 (Power Series).label Let $E, F$ be locally convex spaces $K \in \RC$ with $F$ being complete, $\bracsn{T_n}_{0}^{\infty}$ with $T_{n} \in L^{n}(E; F)$ for each $n \in \natz$, and $a \in E$, then the power series of $\bracsn{T_n}_{0}^{\infty}$ about $a$ is the function

defined on points on which the series converges.

Definition 30.7.2 (Radius of Convergence).label Let $E$ be a normed space over $K \in \RC$, $F$ be a complete locally convex space over $K$, and $\sum_{n = 0}^{\infty} T_{n}(x - a)^{(n)}$ be a power series about $a \in E$, $\rho: F \to [0, \infty)$ be a continuous seminorm on $F$. For each $T \in L^{n}(E; F)$, let

then $R_{\rho} \in [0, \infty]$ be defined by^[1]

is the radius of convergence of $\sum_{n = 0}^{\infty} T_{n}(x - a)^{(n)}$ with respect to $\rho$, and

1. (1)

   For each $0 < r < R$, the series converges uniformly and absolutely on $B_{E}(a, r)$ with respect to $\rho$.

2. (2)

   Let

   \[R = \inf\bracs{R_\rho| \rho: F \to [0, \infty) \text{ is a continuous seminorm}}\]

   
   

   the series converges uniformly and absolutely on $B_{E}(a, R)$, and $R$ is the radius of convergence of $\sum_{n = 0}^{\infty} T_{n}(x - a)^{(n)}$.

Remark 30.7.1.label In Definition 30.7.2, the radius of convergence appears to be an arbitrary lower bound on the domain of convergence. However, in the more specialised case of power series from $\complex$ to $\complex$ or in a Banach algebra, $R$ is the largest constant such that the series converges uniformly and absolutely on all $B(0, r)$ where $0 < r < R$. The lack of this ”maximum” claim is why the above statement is a definition.

Theorem 30.7.3 (Termwise Differentiation).label Let $E$ be a normed space over $K \in \RC$, $F$ be a complete locally convex space over $K$, $f(x) = \sum_{n = 0}^{\infty} T_{n}(x - a)^{(n)}$ a power series about $a \in E$, and $R$ be its radius of convergence, then

1. (1)

   $f \in C^{\infty}(B(a, R); F)$ is infinitely Fréchet differentiable.

2. (2)

   For each $x \in B(a, R)$ and $h \in E$,

   \[Df(x)(h) = \sum_{n = 0}^{\infty} \sum_{k = 1}^{n+1}T_{n+1}(((x-a)^{(n)}, h)^{\bracs{k}})\]

   
   
3. (3)

   The radius of convergence of the above series is at least $R$.