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# 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 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. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.

Related:

AscoliArzelaTheorem, SpaceOfAnalyticFunctions

Type of Math Object:

Theorem

Major Section:

Reference

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## Mathematics Subject Classification

30C99*no label found*

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new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

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new correction: Misspelled name by DavidSteinsaltz

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new image: plot W(t) = P(waiting time <= t) by robert_dodier