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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 Reflexive space ： ウィキペディア英語版 Reflexive space In the area of mathematics known as functional analysis, a reflexive space is a Banach space (or more generally a locally convex topological vector space) that coincides with the continuous dual of its continuous dual space, both as linear space and as topological space. Reflexive Banach spaces are often characterized by their geometric properties. == Reflexive Banach spaces == Suppose $X$ is a normed vector space over the number field $\backslash mathbb\; =\; \backslash mathbb$ or $\backslash mathbb\; =\; \backslash mathbb$ (the real or complex numbers), with a norm $\backslash |\backslash cdot\backslash |$. Consider its dual normed space $X\text{'}$, that consists of all continuous linear functionals $f:X\backslash to$ and is equipped with the dual norm $\backslash |\backslash cdot\backslash |\text{'}$ defined by :$\backslash |f\backslash |\text{'}\; =\; \backslash sup\; \backslash .$ The dual $X\text{'}$ is a normed space (a Banach space to be precise), and its dual normed space $X\text{'}\text{'}=(X\text{'})\text{'}$ is called bidual space for $X$. The bidual consists of all continuous linear functionals $h:X\text{'}\backslash to$ and is equipped with the norm $\backslash |\backslash cdot\backslash |\text{'}\text{'}$ dual to $\backslash |\backslash cdot\backslash |\text{'}$. Each vector $x\backslash in\; X$ generates a scalar function $J(x):X\text{'}\backslash to$ by the formula: :$$ J(x)(f)=f(x),\qquad f\in X', and $J(x)$ is a continuous linear functional on $X\text{'}$, ''i.e.'', $J(x)\backslash in\; X\text{'}\text{'}$. One obtains in this way a map :$J:\; X\; \backslash to\; X\text{'}\text{'}$ called evaluation map, that is linear. It follows from the Hahn–Banach theorem that $J$ is injective and preserves norms: :$$ \forall x\in X\qquad \|J(x)\|''=\|x\|, ''i.e.'', $J$ maps $X$ isometrically onto its image $J(X)$ in $X\text{'}\text{'}$. Furthermore, the image $J(X)$ is closed in $X\text{'}\text{'}$, but it need not be equal to $X\text{'}\text{'}$. A normed space $X$ is called reflexive if it satisfies the following equivalent conditions: :(i) the evaluation map $J:X\backslash to\; X\text{'}\text{'}$ is surjective, :(ii) the evaluation map $J:X\backslash to\; X\text{'}\text{'}$ is an isometric isomorphism of normed spaces, :(iii) the evaluation map $J:X\backslash to\; X\text{'}\text{'}$ is an isomorphism of normed spaces. A reflexive space $X$ is a Banach space, since $X$ is then isometric to the Banach space $X\text{'}\text{'}$. 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「Reflexive 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.