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)
For each $0 < r < R$, the series converges uniformly and absolutely on $B_{E}(a, r)$ with respect to $\rho$.
- (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 $\sum_{n = 0}^{\infty} T_{n}(x - a)^{(n)}$.
Proof. For all $x \in B_{E}(a, r)$,
For any $s \in (r, R)$, there exists $N \in \natp$ such that $\norm{T_n}_{L^n(E; F)}^{1/n}\le 1/s$ for all $n \ge N$. In which case,
As this estimate holds uniformly on $B_{E}(a, r)$, the series converges uniformly and absolutely on $B_{E}(a, r)$ with respect to $\rho$.$\square$
- Under the abuse that $1/\infty = 0$ and $1/0 =\infty$.keyboard_return
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