Theorem 26.6.3 (Termwise Differentiation).label Let $E$ be a normed space over $K \in \RC$, $F$ be a Banach 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)
$f \in C^{\infty}(B(a, R); F)$.
- (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): For each $n \in \natz$, let
then $\norm{S_n}_{L^n(E; L(E; F))}\le (n+1)\norm{T_{n+1}}_{L^{n+1}(E; F)}$. Since $(n+1)^{1/n}$ is convergent and $\{||T_{n}||_{L^n(E; F)}^{1/n}\}$ is bounded,
so the radius of convergence of the proposed series is at least $R$.
(2): By the Theorem 26.4.6, for each $N \in \natp$,
By Definition 26.6.2, the proposed series converges uniformly on $B(a, r)$ for each $0 < r < R$. Thus by Theorem 26.2.8, $f$ is differentiable on $B(a, R)$ with
(1): By (2), (3) applied inductively to $D^{n}f$.$\square$