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Montel's theorem (Theorem)

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

Theorem 1 (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.

Bibliography

1
John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.



"Montel's theorem" is owned by jirka.
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See Also: Ascoli-Arzelà theorem, space of analytic functions

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Cross-references: compact subsets, converges uniformly, subsequence, sequence, locally bounded, holomorphic functions, region
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This is version 5 of Montel's theorem, born on 2004-04-11, modified 2005-03-07.
Object id is 5754, canonical name is MontelsTheorem.
Accessed 3448 times total.

Classification:
AMS MSC30C99 (Functions of a complex variable :: Geometric function theory :: Miscellaneous)

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