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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 Bessel's inequality ： ウィキペディア英語版 Bessel's inequality In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element $x$ in a Hilbert space with respect to an orthonormal sequence. Let $H$ be a Hilbert space, and suppose that $e\_1,\; e\_2,\; ...$ is an orthonormal sequence in $H$. Then, for any $x$ in $H$ one has :$\backslash sum\_^\backslash left\backslash vert\backslash left\backslash langle\; x,e\_k\backslash right\backslash rangle\; \backslash right\backslash vert^2\; \backslash le\; \backslash left\backslash Vert\; x\backslash right\backslash Vert^2$ where 〈•,•〉 denotes the inner product in the Hilbert space $H$. If we define the infinite sum :$x\text{'}\; =\; \backslash sum\_^\backslash left\backslash langle\; x,e\_k\backslash right\backslash rangle\; e\_k,$ consisting of 'infinite sum' of vector resolute $x$ in direction $e\_k$, Bessel's inequality tells us that this series converges. One can think of it that there exists $x\text{'}\; \backslash in\; H$ which can be described in terms of potential basis $e\_1,\; e\_2,\; ...$. For a complete orthonormal sequence (that is, for an orthonormal sequence which is a basis), we have Parseval's identity, which replaces the inequality with an equality (and consequently $x\text{'}$ with $x$). Bessel's inequality follows from the identity: :$0\; \backslash le\; \backslash left\backslash |\; x\; -\; \backslash sum\_^n\; \backslash langle\; x,\; e\_k\; \backslash rangle\; e\_k\backslash right\backslash |^2\; =\; \backslash |x\backslash |^2\; -\; 2\; \backslash sum\_^n\; |\backslash langle\; x,\; e\_k\; \backslash rangle\; |^2\; +\; \backslash sum\_^n\; |\; \backslash langle\; x,\; e\_k\; \backslash rangle\; |^2\; =\; \backslash |x\backslash |^2\; -\; \backslash sum\_^n\; |\; \backslash langle\; x,\; e\_k\; \backslash rangle\; |^2,$ which holds for any natural ''n''. ==See also== * Cauchy–Schwarz inequality 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「Bessel's 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.