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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 Clarkson's inequalities ： ウィキペディア英語版 Clarkson's inequalities In mathematics, Clarkson's inequalities, named after James A. Clarkson, are results in the theory of ''L''''p'' spaces. They give bounds for the ''L''''p''-norms of the sum and difference of two measurable functions in ''L''''p'' in terms of the ''L''''p''-norms of those functions individually. ==Statement of the inequalities== Let (''X'', Σ, ''μ'') be a measure space; let ''f'', ''g'' : ''X'' → R be measurable functions in ''L''''p''. Then, for 2 ≤ ''p'' < +∞, :$\backslash left\backslash |\; \backslash frac\; \backslash right\backslash |\_^p\; +\; \backslash left\backslash |\; \backslash frac\; \backslash right\backslash |\_^p\; \backslash le\; \backslash frac\; \backslash left(\; \backslash |\; f\; \backslash |\_^p\; +\; \backslash |\; g\; \backslash |\_^p\; \backslash right).$ For 1 < ''p'' < 2, :$\backslash left\backslash |\; \backslash frac\; \backslash right\backslash |\_^q\; +\; \backslash left\backslash |\; \backslash frac\; \backslash right\backslash |\_^q\; \backslash le\; \backslash left(\; \backslash frac\; \backslash |\; f\; \backslash |\_^p\; +\backslash frac\; \backslash |\; g\; \backslash |\_^p\; \backslash right)^\backslash frac,$ where :$\backslash frac1\; +\; \backslash frac1\; =\; 1,$ i.e., ''q'' = ''p'' ⁄ (''p'' − 1). The case ''p'' ≥ 2 is somewhat easier to prove, being a simple application of the triangle inequality and the convexity of :$x\; \backslash mapsto\; x^p.\; \backslash ,$ 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「Clarkson's inequalities」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.