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)
$f \in C^{\infty}(B(a, R); F)$ is infinitely Fréchet differentiable.
- (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)
The radius of convergence of the above series is at least $R$.
Proof. (3): Let $\rho: F \to [0, \infty)$ be a continuous seminorm. For each $n \in \natz$ and $T \in L^{n}(E; F)$, let
and
then $[S_{n}]_{L^n(E; L(E; F)), \rho}\le (n+1)[T_{n+1}]_{L^{n+1}(E; F), \rho}$. Since $(n+1)^{1/n}$ is convergent and $\{[T_{n+1}]_{L^{n+1}(E; F), \rho}\}_{1}^{\infty}$ is bounded,
so the radius of convergence of the proposed series is at least $R$.
(2): By the Theorem 30.4.7, for each $N \in \natp$,
By Definition 30.7.2, the proposed series converges uniformly on $B(a, r)$ for each $0 < r < R$. Thus by Theorem 30.2.8, $f$ is differentiable on $B(a, R)$ with
(1): By (2), (3) applied inductively to $D^{n}f$.$\square$
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