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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 Normed vector space ： ウィキペディア英語版 Normed vector space In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space R''n''. The following properties of "vector length" are crucial. # The zero vector, 0, has zero length; every other vector has a positive length. #: $\backslash |x\backslash |>\; 0$ if $x\backslash ne0$ # Multiplying a vector by a positive number changes its length without changing its direction. Moreover, #: $\backslash |\backslash alpha\; x\backslash |=|\backslash alpha|\; \backslash |x\backslash |$ for any scalar $\backslash alpha.$ # The triangle inequality holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line. #: $\backslash |x+y\backslash |\; \backslash le\; \backslash |x\backslash |+\backslash |y\backslash |$ for any vectors x and y. (triangle inequality) The generalization of these three properties to more abstract vector spaces leads to the notion of norm. A vector space on which a norm is defined is then called a normed vector space. Normed vector spaces are central to the study of linear algebra and functional analysis. ==Definition== A normed vector space is a pair (''V'', ‖·‖ ) where ''V'' is a vector space and ‖·‖ a norm on ''V''. A seminormed vector space is a pair (''V'',''p'') where ''V'' is a vector space and ''p'' a seminorm on ''V''. We often omit ''p'' or ‖·‖ and just write ''V'' for a space if it is clear from the context what (semi) norm we are using. In a more general sense, a vector norm can be taken to be any real-valued function that satisfies these three properties. The properties 1. and 2. together imply that :$\backslash |x\backslash |=\; 0$ if and only if $x=0$. A useful variation of the triangle inequality is :$\backslash |x-y\backslash |\; \backslash ge\; |\; \backslash |x\backslash |-\backslash |y\backslash |\; |$ for any vectors x and y. This also shows that a vector norm is a continuous function. Note that property 2 depends on a choice of norm $|\backslash alpha|$ on the field of scalars. When the scalar field is $\backslash mathbb\; R$ (or more generally a subset of $\backslash mathbb\; C$), this is usually taken to be the ordinary absolute value, but other choices are possible. For example, for a vector space over $\backslash mathbb\; Q$ one could take $|\backslash alpha|$ to be the ''p''-adic norm, which gives rise to a different class of normed vector spaces. 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「Normed vector space」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.