Normal family theorem: Difference between revisions
(New page: {{basic fact}} ==Statement== Suppose <math>U \subset \mathbb{C}</math> is an open subset and <math>f_\alpha: U \to \mathbb{C}</math> is a family of holomorphic functions such that th...) |
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==Statement== | ==Statement== | ||
Suppose <math>U \subset \mathbb{C}</math> is an open subset and <math>f_\alpha: U \to \mathbb{C}</math> is a family of [[holomorphic function]]s such that the <math>f_\alpha</math> form a [[normal family]]: for every compact subset <math>K \subset U</math>, the restrictions of <math>f_\alpha</math> to <math>K</math> are uniformly bounded on <math>K</math>. Then, there exists a subsequence of the <math>f_\alpha</math>s, say <math>f_n</math>, such that <math>f_n</math> converge to a holomorphic function, and the convergence is uniform on compact subsets. | Suppose <math>U \subset \mathbb{C}</math> is an open subset and <math>f_\alpha: U \to \mathbb{C}</math> is a family of [[holomorphic function]]s such that the <math>f_\alpha</math> form a [[normal family]]: for every compact subset <math>K \subset U</math>, the restrictions of <math>f_\alpha</math> to <math>K</math> are uniformly bounded on <math>K</math>. Then, there exists a subsequence of the <math>f_\alpha</math>s, say <math>f_n</math>, such that <math>f_n</math> converge to a holomorphic function, and the convergence is uniform on compact subsets. | ||
Latest revision as of 19:17, 18 May 2008
This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis
This fact is an application of the following pivotal fact/result/idea: Arzela-Ascoli theorem
View other applications of Arzela-Ascoli theorem OR Read a survey article on applying Arzela-Ascoli theorem
Statement
Suppose is an open subset and is a family of holomorphic functions such that the form a normal family: for every compact subset , the restrictions of to are uniformly bounded on . Then, there exists a subsequence of the s, say , such that converge to a holomorphic function, and the convergence is uniform on compact subsets.
