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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 Grothendieck inequality ： ウィキペディア英語版 Grothendieck inequality In mathematics, the Grothendieck inequality states that there is a universal constant ''k'' with the following property. If ''a''''i'',''j'' is an ''n'' by ''n'' (real or complex) matrix with :$\backslash left|\; \backslash sum\_\; a\_\; s\_i\; t\_j\; \backslash right|\backslash le\; 1$ for all (real or complex) numbers ''s''''i'', ''t''''j'' of absolute value at most 1, then :$\backslash left|\; \backslash sum\_\; a\_\; \backslash langle\; S\_i\; ,\; T\_j\; \backslash rangle\; \backslash right|\backslash le\; k$, for all vectors ''S''''i'', ''T''''j'' in the unit ball ''B''(''H'') of a (real or complex) Hilbert space ''H''. The smallest constant ''k'' which satisfies this property for all ''n'' by ''n'' matrices is called a Grothendieck constant and denoted ''k''(''n''). In fact there are two Grothendieck constants ''k''R(''n'') and ''k''C(''n'') for each ''n'' depending on whether one works with real or complex numbers, respectively.〔.〕 The Grothendieck inequality and Grothendieck constants are named after Alexander Grothendieck, who proved the inequality and the existence of the constants in a paper published in 1953.〔 ==Bounds on the constants== The sequences ''k''R(''n'') and ''k''C(''n'') are easily seen to be increasing, and Grothendieck's result states that they are bounded, so they have limits. With ''k''R defined to be sup''n'' ''k''R(''n'') then Grothendieck proved that: $1.57\; \backslash approx\; \backslash frac\; \backslash leq\; k\_\; \backslash leq\; \backslash mathrm(\backslash frac)\; \backslash approx\; 2.3$. improved the result by proving: 1.67696... ≤ ''k''R ≤ 1.7822139781...=$\backslash frac$, conjecturing that the upper bound is tight. However, this conjecture was disproved by . 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「Grothendieck inequality」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.