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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 Banach–Mazur compactum ： ウィキペディア英語版 Banach–Mazur compactum In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set ''Q''(''n'') of ''n''-dimensional normed spaces. If ''X'' and ''Y'' are two finite-dimensional normed spaces with the same dimension, let GL(''X'',''Y'') denote the collection of all linear isomorphisms ''T'' : ''X'' → ''Y''. The Banach–Mazur distance between ''X'' and ''Y'' is defined by : $\backslash delta(X,\; Y)\; =\; \backslash log\; \backslash Bigl(\; \backslash inf\; \backslash (X,\; Y)\; \backslash \}\; \backslash Bigr).$ Equipped with the metric ''δ'', the space ''Q''(''n'') is a compact metric space, called the Banach–Mazur compactum. Many authors prefer to work with the ''multiplicative Banach–Mazur distance'' :$d(X,\; Y)\; :=\; \backslash mathrm^\; =\; \backslash inf\; \backslash (X,\; Y)\; \backslash \},$ for which ''d''(''X'', ''Z'') ≤ ''d''(''X'', ''Y'') ''d''(''Y'', ''Z'') and ''d''(''X'', ''X'') = 1. F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate: : $d(X,\; \backslash ell^n\_2)\; \backslash le\; \backslash sqrt,\; \backslash ,$ 〔http://users.uoa.gr/~apgiannop/cube.ps〕 where ℓ''n''2 denotes R''n'' with the Euclidean norm (see the article on ''L''''p'' spaces). From this it follows that ''d''(''X'', ''Y'') ≤ ''n'' for every couple (''X'', ''Y'') in ''Q''(''n''). However, for the classical spaces, this upper bound for the diameter of ''Q''(''n'') is far from being approached. For example, the distance between ℓ''n''1 and ℓ''n''∞ is (only) of order ''n''1/2 (up to a multiplicative constant independent from the dimension ''n''). A major achievement in the direction of estimating the diameter of ''Q''(''n'') is due to E. Gluskin, who proved in 1981 that the (multiplicative) diameter of the Banach–Mazur compactum is bounded below by ''c'' ''n'', for some universal ''c'' > 0. Gluskin's method introduces a class of random symmetric polytopes ''P''(''ω'') in R''n'', and the normed spaces ''X''(''ω'') having ''P''(''ω'') as unit ball (the vector space is R''n'' and the norm is the gauge of ''P''(''ω'')). The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space ''X''(''ω''). ''Q''(2) is an absolute extensor.〔(The Banach–Mazur compactum is not homeomorphic to the Hilbert cube )〕 On the other hand, ''Q''(2) is not homeomorphic to a Hilbert cube. ==Notes== 〔 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「Banach–Mazur compactum」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.