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
Canonical name	MontelsTheorem
Date of creation	2013-03-22 14:17:52
Last modified on	2013-03-22 14:17:52
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	8
Author	jirka (4157)
Entry type	Theorem
Classification	msc 30C99
Related topic	AscoliArzelaTheorem
Related topic	SpaceOfAnalyticFunctions