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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Uniformly Cauchy sequence ： ウィキペディア英語版 Uniformly Cauchy sequence In mathematics, a sequence of functions $\backslash$ from a set ''S'' to a metric space ''M'' is said to be uniformly Cauchy if: * For all $\backslash varepsilon\; >\; 0$, there exists $N>0$ such that for all $x\backslash in\; S$: whenever $m,\; n\; >\; N$. Another way of saying this is that $d\_u\; (f\_,\; f\_)\; \backslash to\; 0$ as $m,\; n\; \backslash to\; \backslash infty$, where the uniform distance $d\_u$ between two functions is defined by :$d\_\; (f,\; g)\; :=\; \backslash sup\_\; d\; (f(x),\; g(x)).$ == Convergence criteria == A sequence of functions from ''S'' to ''M'' is pointwise Cauchy if, for each ''x'' ∈ ''S'', the sequence is a Cauchy sequence in ''M''. This is a weaker condition than being uniformly Cauchy. In general a sequence can be pointwise Cauchy and not pointwise convergent, or it can be uniformly Cauchy and not uniformly convergent. Nevertheless, if the metric space ''M'' is complete, then any pointwise Cauchy sequence converges pointwise to a function from ''S'' to ''M''. Similarly, any uniformly Cauchy sequence will tend uniformly to such a function. The uniform Cauchy property is frequently used when the ''S'' is not just a set, but a topological space, and ''M'' is a complete metric space. The following theorem holds: * Let ''S'' be a topological space and ''M'' a complete metric space. Then any uniformly Cauchy sequence of continuous functions ''f''n : ''S'' → ''M'' tends uniformly to a unique continuous function ''f'' : ''S'' → ''M''. 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「Uniformly Cauchy sequence」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.