Montel's theoremMontelsTheorem2004-04-11 22:08:012005-03-07 20:05:55Theorem% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{defn}{Definition}Suppose that $G \subset {\mathbb{C}}$ is a region.
\begin{thm}[Montel]
A set ${\mathcal{F}}$ of holomorphic functions $f\colon G \to {\mathbb{C}}$ is \PMlinkname{normal}{NormalFamily} if and only if ${\mathcal{F}}$ is
locally bounded.
\end{thm}
In other words a sequence of holomorphic functions $\{ f_n \}$ has a subsequence which converges uniformly
on compact subsets to a holomorphic function $f \colon G \to {\mathbb{C}}$
if and only if the set $\{ f_n \}$ is locally bounded.
\begin{thebibliography}{9}
\bibitem{Conway:complexI}
John~B. Conway.
{\em \PMlinkescapetext{Functions of One Complex Variable I}}.
Springer-Verlag, New York, New York, 1978.
\end{thebibliography}