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 ${\\mathcal{F}}$ islocally bounded.

In other words a sequence of holomorphic functions $\\{f_{n}\\}$ has a subsequence which converges uniformlyon 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.

单词	MontelsTheorem
释义	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 ${\\mathcal{F}}$ islocally bounded. In other words a sequence of holomorphic functions $\\{f_{n}\\}$ has a subsequence which converges uniformlyon 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.
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