Proposition 27.4.2.label Let $E$ be a complete separated locally convex space over $\complex$, and $f \in H(\complex; E)$, then for any $z_{0} \in \complex$ and $z \in \complex$,
\[f(z) = \sum_{k = 0}^{\infty} \frac{1}{k!}D^{k}f(z_{0})(z - z_{0})^{k}\]
where the radius of convergence of the series is infinite.
Proof. See (4) of Definition 27.1.6.$\square$
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