翻訳と辞書 Words near each other ・ "O" Is for Outlaw ・ "O"-Jung.Ban.Hap. ・ "Ode-to-Napoleon" hexachord ・ "Oh Yeah!" Live ・ "Our Contemporary" regional art exhibition (Leningrad, 1975) ・ "P" Is for Peril ・ "Pimpernel" Smith ・ "Polish death camp" controversy ・ "Pro knigi" ("About books") ・ "Prosopa" Greek Television Awards ・ "Pussy Cats" Starring the Walkmen ・ "Q" Is for Quarry ・ "R" Is for Ricochet ・ "R" The King (2016 film) ・ "Rags" Ragland ・ ! (album) ・ ! (disambiguation) ・ !! ・ !!! ・ !!! (album) ・ !!Destroy-Oh-Boy!! ・ !Action Pact! ・ !Arriba! La Pachanga ・ !Hero ・ !Hero (album) ・ !Kung language ・ !Oka Tokat ・ !PAUS3 ・ !T.O.O.H.! ・ !Women Art Revolution Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

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.