Corollary 27.1.5 (Cauchy’s Estimate).label Let $E$ be a complete separated locally convex space over $\complex$, $U \subset \complex$ be open, $z_{0} \in U$, $r > 0$ such that $\ol{B(z_0, r)}\subset U$, then for any $k \in \natz$ and continuous seminorm $[\cdot]_{E}: E \to [0, \infty)$,
\[[D^{k}f(z_{0})]_{E} \le \frac{k!}{r^{k}}\sup_{z \in \ol{B(z_0, r)}}[f(z)]_{E}\]
Proof. By Cauchy’s Integral Formula and Proposition 13.4.1,
\begin{align*}D^{k}f(z_{0})&= \frac{k!}{2\pi i}\int_{\omega_{z_0, r}}\frac{f(z)}{(z - z_{0})^{k+1}}dz \\ [D^{k}f(z_{0})]_{E}&\le \frac{k!}{2\pi i}\int_{0}^{2\pi}\frac{[f(z)]_{E}}{|z - z_{0}|^{k+1}}dz \\&= \frac{k!}{2\pi i}\int_{0}^{2\pi}\frac{[f(z)]_{E}}{r^{k+1}}dz \le \frac{k!}{r^{k}}\sup_{z \in \ol{B(z_0, r)}}[f(z)]_{E}\end{align*}
$\square$
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