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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 Kuratowski embedding ： ウィキペディア英語版 Kuratowski embedding In mathematics, the Kuratowski embedding allows one to view any metric space as a subset of some Banach space. It is named after Kazimierz Kuratowski. Specifically, if (''X'',''d'') is a metric space, ''x''0 is a point in ''X'', and ''Cb''(''X'') denotes the Banach space of all bounded continuous real valued functions on ''X'' with the supremum norm, then the map :$\backslash Phi\; :\; X\; \backslash rarr\; C\_b(X)$ defined by :$\backslash Phi(x)(y)\; =\; d(x,y)-d(x\_0,y)\; \backslash quad\backslash mbox\backslash quad\; x,y\backslash in\; X$ is an isometry. Note that this embedding depends on the chosen point ''x''0 and is therefore not entirely canonical. The Kuratowski–Wojdysławski theorem states that every bounded metric space ''X'' is isometric to a closed subset of a convex subset of some Banach space.〔. Theorem III.8.1〕 (N.B. the image of this embedding is closed in the convex subset, not necessarily in the Banach space.) Here we use the isometry :$\backslash Psi\; :\; X\; \backslash rarr\; C\_b(X)$ defined by :$\backslash Psi(x)(y)\; =\; d(x,y)\; \backslash quad\backslash mbox\backslash quad\; x,y\backslash in\; X$ The convex set mentioned above is the convex hull of Ψ(''X''). In both of these embedding theorems, we may replace ''Cb''(''X'') by the Banach space ''ℓ'' ∞(''X'') of all bounded functions ''X'' → R, again with the supremum norm, since ''Cb''(''X'') is a closed linear subspace of ''ℓ'' ∞(''X''). These embedding results are useful because Banach spaces have a number of useful properties not shared by all metric spaces: they are vector spaces which allows one to add points and do elementary geometry involving lines and planes etc.; and they are complete. Given a function with codomain ''X'', it is frequently desirable to extend this function to a larger domain, and this often requires simultaneously enlarging the codomain to a Banach space containing ''X''. == History == Formally speaking, this embedding was first introduced by Kuratowski,〔Kuratowski, C. (1935) "Quelques problèmes concernant les espaces métriques non-separables" (Some problems concerning non-separable metric spaces), Fundamenta Mathematicae 25: pp. 534-545.〕 but a very close variation of this embedding appears already in the paper of Fréchet〔''Fréchet M.'' (1906) "Sur quelques points du calcul fonctionnel", Rendiconti del Circolo Matematico di Palermo 22: 1—74.〕 where he first introduces the notion of metric 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「Kuratowski embedding」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.