Montel's theorem

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In complex analysis, an area of mathematics, Montel's theorem, named after Paul Montel, is a theorem about sets (also called families) of holomorphic functions.

[edit] Montel's theorem

Given a family F of holomorphic functions defined on an open subset D of the complex numbers C, then F is a normal family if and only if F is locally bounded.

The key part of this theorem can be reformulated as follows. Any locally bounded sequence of holomorphic functions f_n defined on D has a subsequence which converges compactly to a holomorphic function f.

[edit] References

• John B. Conway (1978). Functions of One Complex Variable I. Springer-Verlag. ISBN 0-387-90328-3. 
• J. L. Schiff (1993). Normal Families. Springer-Verlag. ISBN 0-387-97967-0. 

[edit] See also

• Arzelà-Ascoli theorem
• Montel space

This article incorporates material from Montel's theorem on PlanetMath, which is licensed under the GFDL.