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In mathematics, with special application to complex analysis, a normal family is a pre-compact family of continuous functions. Informally, this means that the functions in the family are not widely spread out, but rather stick together in a somewhat "clustered" manner. It is of general interest to understand compact sets in function spaces, since these are usually truly infinite-dimensional in nature. More formally, a family (equivalently, a set) ''F'' of continuous functions ''f'' defined on some complete metric space ''X'' with values in another complete metric space ''Y'' is called normal if every sequence of functions in ''F'' contains a subsequence which converges uniformly on compact subsets of ''X'' to a continuous function from ''X'' to ''Y''. That is, for every sequence of functions in ''F'', there is a subsequence and a continuous function from ''X'' to ''Y'' such that the following holds for every compact subset ''K'' contained in ''X'': where is the distance metric associated with the complete metric space ''Y''. ==Complex analysis== This definition is often used in complex analysis for spaces of holomorphic functions. In this case, sets ''X'' and ''Y'' are regions in the complex plane, and . As a consequence of Cauchy's integral theorem, a sequence of holomorphic functions that converges uniformly on compact sets must converge to a holomorphic function. Thus in complex analysis a normal family ''F'' of holomorphic functions in a region ''X'' of the complex plane with values in ''Y'' = C is such that every sequence in ''F'' contains a subsequence which converges uniformly on compact subsets of ''X'' to a holomorphic function. Montel's theorem asserts that every locally bounded family of holomorphic functions is normal. Another space where this is often used is the space of meromorphic functions. This is similar to the holomorphic case, but instead of using the standard metric (distance) for convergence we must use the spherical metric. That is if ''d'' is the spherical metric, then want : compactly to mean that : goes to 0 uniformly on compact subsets. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「normal family」の詳細全文を読む スポンサード リンク
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