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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 Hanner's inequalities ： ウィキペディア英語版 Hanner's inequalities In mathematics, Hanner's inequalities are results in the theory of ''L''''p'' spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of ''L''''p'' spaces for ''p'' ∈ (1, +∞) than the approach proposed by James A. Clarkson in 1936. ==Statement of the inequalities== Let ''f'', ''g'' ∈ ''L''''p''(''E''), where ''E'' is any measure space. If ''p'' ∈ (), then :$\backslash |f+g\backslash |\_p^p\; +\; \backslash |f-g\backslash |\_p^p\; \backslash geq\; \backslash big(\; \backslash |f\backslash |\_p\; +\; \backslash |g\backslash |\_p\; \backslash big)^p\; +\; \backslash big|\; \backslash |f\backslash |\_p-\backslash |g\backslash |\_p\; \backslash big|^p.$ The substitutions ''F'' = ''f'' + ''g'' and ''G'' = ''f'' − ''g'' yield the second of Hanner's inequalities: :$2^p\; \backslash big(\; \backslash |F\backslash |\_p^p\; +\; \backslash |G\backslash |\_p^p\; \backslash big)\; \backslash geq\; \backslash big(\; \backslash |F+G\backslash |\_p\; +\; \backslash |F-G\backslash |\_p\; \backslash big)^p\; +\; \backslash big|\; \backslash |F+G\backslash |\_p-\backslash |F-G\backslash |\_p\; \backslash big|^p.$ For ''p'' ∈ [2, +∞) the inequalities are reversed (they remain non-strict). Note that for ''p'' = 2 the inequalities become equalities, and the second yields the parallelogram rule. 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「Hanner'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.