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The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis. The notion of equicontinuity was introduced at around the same time by and . A weak form of the theorem was proven by , who established the sufficient condition for compactness, and by , who established the necessary condition and gave the first clear presentation of the result. A further generalization of the theorem was proven by , to sets of real-valued continuous functions with domain a compact metric space . Modern formulations of the theorem allow for the domain to be compact Hausdorff and for the range to be an arbitrary metric space. More general formulations of the theorem exist that give necessary and sufficient conditions for a family of functions from a compactly generated Hausdorff space into a uniform space to be compact in the compact-open topology. . == Statement and first consequences == A sequence of continuous functions on an interval is uniformly bounded if there is a number ''M'' such that : for every function belonging to the sequence, and every . The sequence is ''equicontinuous'' if, for every , there exists such that : whenever for all functions in the sequence. Succinctly, a sequence is equicontinuous if and only if all of its elements admit ''the same'' modulus of continuity. In simplest terms, the theorem can be stated as follows: :Consider a sequence of real-valued continuous functions defined on a closed and bounded interval of the real line. If this sequence is uniformly bounded and equicontinuous, then there exists a subsequence (''fnk'') that converges uniformly. The converse is also true, in the sense that if every subsequence of itself has a uniformly convergent subsequence, then is uniformly bounded and equicontinuous. (See below for a proof.) 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「Arzelà–Ascoli theorem」の詳細全文を読む スポンサード リンク
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