Theorem 27.4.3 (Liouville).label Let $E$ be a complete separated locally convex space over $\complex$ and $f \in H(\complex; E)$. If $f$ is bounded, then $f$ is constant.
Proof. Let $x_{0} \in \complex$ and $[\cdot]_{E}: [E, \infty)$ be a continuous seminorm on $E$, then by Cauchy’s Estimate,
\[[Df(z_{0})]_{E} \le \frac{1}{r}\sup_{z \in \ol{B(z_0, r)}}[f(z)]_{E}\]
for all $r > 0$. Therefore $Df = 0$, and $f$ is constant.$\square$
Post a Comment