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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 Quotient of subspace theorem ： ウィキペディア英語版 Quotient of subspace theorem In mathematics, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman.〔The original proof appeared in . See also .〕 Let (''X'', ||·||) be an ''N''-dimensional normed space. There exist subspaces ''Z'' ⊂ ''Y'' ⊂ ''X'' such that the following holds: * The quotient space ''E'' = ''Y'' / ''Z'' is of dimension dim E ≥ ''c'' ''N'', where ''c'' > 0 is a universal constant. * The induced norm || · || on ''E'', defined by :: $\backslash |\; e\; \backslash |\; =\backslash min\_\; \backslash |\; y\; \backslash |,\; \backslash quad\; e\; \backslash in\; E,$ is uniformly isomorphic to Euclidean. That is, there exists a positive quadratic form ("Euclidean structure") ''Q'' on ''E'', such that :: $\backslash frac\; \backslash leq\; \backslash |\; e\; \backslash |\; \backslash leq\; K\; \backslash sqrt$ for $e\; \backslash in\; E,$ with ''K'' > 1 a universal constant. The statement is relative easy to prove by induction on the dimension of ''Z'' (even for ''Y=Z'', ''X''=''0'', ''c=1'') with a ''K'' that depends only on ''N''; the point of the theorem is that ''K'' is independent of ''N''. In fact, the constant ''c'' can be made arbitrarily close to 1, at the expense of the constant ''K'' becoming large. The original proof allowed :$c(K)\; \backslash approx\; 1\; -\; \backslash text\; /\; \backslash log\; \backslash log\; K.$〔See references for improved estimates.〕 ==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「Quotient of subspace theorem」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.