# Montel’s theorem

Suppose that $G\subset{\mathbb{C}}$ is a region.

###### Theorem (Montel).

A set ${\mathcal{F}}$ of holomorphic functions $f\colon G\to{\mathbb{C}}$ is normal (http://planetmath.org/NormalFamily) if and only if ${\mathcal{F}}$ is locally bounded.

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.

## References

• 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
Title Montel’s theorem MontelsTheorem 2013-03-22 14:17:52 2013-03-22 14:17:52 jirka (4157) jirka (4157) 8 jirka (4157) Theorem msc 30C99 AscoliArzelaTheorem SpaceOfAnalyticFunctions