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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 Uniform norm ： ウィキペディア英語版 Uniform norm In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions ''f'' defined on a set ''S'' the non-negative number :$\backslash |f\backslash |\_\backslash infty=\backslash |f\backslash |\_=\backslash sup\backslash left\backslash .$ This norm is also called the supremum norm, the Chebyshev norm, or the infinity norm. The name "uniform norm" derives from the fact that a sequence of functions $\backslash$ converges to f under the metric derived from the uniform norm if and only if $f\_n$ converges to $f$ uniformly. If we allow unbounded functions, this formula does not yield a norm or metric in a strict sense, although the obtained so-called extended metric still allows one to define a topology on the function space in question. If ''f'' is a continuous function on a closed interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the maximum norm. In particular, for the case of a vector $x=(x\_1,\backslash dots,x\_n)$ in finite dimensional coordinate space, it takes the form :$\backslash |x\backslash |\_\backslash infty=\backslash max\backslash .$ The reason for the subscript "∞" is that whenever ''f'' is continuous :$\backslash lim\_\backslash |f\backslash |\_p=\backslash |f\backslash |\_\backslash infty,$ where :$\backslash |f\backslash |\_p=\backslash left(\backslash int\_D\; \backslash left|f\backslash right|^p\backslash ,d\backslash mu\backslash right)^$ where ''D'' is the domain of ''f'' (and the integral amounts to a sum if ''D'' is a discrete set). The binary function :$d(f,g)=\backslash |f-g\backslash |\_\backslash infty$ is then a metric on the space of all bounded functions (and, obviously, any of its subsets) on a particular domain. A sequence converges uniformly to a function ''f'' if and only if :$\backslash lim\_\backslash |f\_n-f\backslash |\_\backslash infty=0.\backslash ,$ We can define closed sets and closures of sets with respect to this metric topology; closed sets in the uniform norm are sometimes called ''uniformly closed'' and closures ''uniform closures''. The uniform closure of a set of functions A is the space of all functions that can be approximated by a sequence of uniformly-converging functions on A. For instance, one restatement of the Stone–Weierstrass theorem is that the set of all continuous functions on $()$ is the uniform closure of the set of polynomials on $()$. For complex continuous functions over a compact space, this turns it into a C * algebra. ==See also== *Chebyshev distance *Uniform continuity *Uniform space 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「Uniform norm」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.