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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 List of Banach spaces ： ウィキペディア英語版 List of Banach spaces In the mathematical field of functional analysis, Banach spaces are among the most important objects of study. In other areas of mathematical analysis, most spaces which arise in practice turn out to be Banach spaces as well. == Classical Banach spaces == According to , the classical Banach spaces are those defined by , which is the source for the following table. Here K denotes the field of real numbers or complex numbers and ''I'' is a closed and bounded interval (). The number ''p'' is a real number with , and ''q'' is its Hölder conjugate (also with ), so that the next equation holds: : $\backslash frac+\backslash frac=1\; ,$ and thus : $q=\backslash frac\; .$ The symbol Σ denotes a σ-algebra of sets, and Ξ denotes just an algebra of sets (for spaces only requiring finite additivity, such as the ba space). The symbol μ denotes a positive measure: that is, a real-valued positive set function defined on a σ-algebra which is countably additive. || |- ! ℓnp || ℓnq || || || $\backslash |x\backslash |\_p\; =\; \backslash left(\backslash sum\_^n\; |x\_i|^p\backslash right)^$ || |- ! ℓn∞ | ℓn1 || || || $\backslash |x\backslash |\_\backslash infty\; =\; \backslash max\_\; |x\_i|$ || |- ! ℓp || ℓq || || || $\backslash |x\backslash |\_p\; =\; \backslash left(\backslash sum\_^\backslash infty\; |x\_i|^p\backslash right)^$ || 1 < p < ∞ |- ! ℓ1 || ℓ∞ || || || $\backslash |x\backslash |\_1\; =\; \backslash sum\_^\backslash infty\; |x\_i|$ || |- ! ℓ∞ || ba || || || $\backslash |x\backslash |\_\backslash infty\; =\; \backslash sup\_i\; |x\_i|$ || |- ! ''c'' | ℓ1 || || || $\backslash |x\backslash |\_\backslash infty\; =\; \backslash sup\_i\; |x\_i|$ || |- ! ''c''0 | ℓ1 || || || $\backslash |x\backslash |\_\backslash infty\; =\; \backslash sup\_i\; |x\_i|$ || Isomorphic but not isometric to ''c''. |- ! ''bv'' | ℓ1 + K || || || $\backslash |x\backslash |\_\; =\; |x\_1|\; +\; \backslash sum\_^\backslash infty|x\_-x\_i|$ || |- ! ''bv''0 | ℓ1 || || || $\backslash |x\backslash |\_\; =\; \backslash sum\_^\backslash infty|x\_-x\_i|$ || |- ! ''bs'' | ba || || || $\backslash |x\backslash |\_\; =\; \backslash sup\_n\backslash left|\backslash sum\_^nx\_i\backslash right|$ || Isometrically isomorphic to ℓ∞. |- ! ''cs'' | ℓ1 || || || $\backslash |x\backslash |\_\; =\; \backslash sup\_n\backslash left|\backslash sum\_^nx\_i\backslash right|$ || Isometrically isomorphic to c. |- ! ''B''(''X'', Ξ) || ba(Ξ) || || || $\backslash |f\backslash |\_B\; =\; \backslash sup\_|f(x)|$ || |- ! ''C''(''X'') | ''rca''(''X'') || || || $\backslash |f\backslash |\_\; =\; \backslash sup\_\backslash left|f(x)\backslash right|$ || ''X'' is a compact Hausdorff space. |- ! ba(Ξ) | ? || || || $\backslash |\backslash mu\backslash |\_\; =\; \backslash sup\_\; |\backslash mu|(A)$ (variation of a measure) || |- ! ca(Σ) | ? || || || $\backslash |\backslash mu\backslash |\_\; =\; \backslash sup\_\; |\backslash mu|(A)$ || |- ! rca(Σ) | ? || || || $\backslash |\backslash mu\backslash |\_\; =\; \backslash sup\_\; |\backslash mu|(A)$ || |- ! Lp(μ) | Lq(μ) || || || $\backslash |f\backslash |\_p\; =\; \backslash left\backslash ^$ || 1 < p < ∞ |- ! L1(μ) | L∞(μ) || || ? || $\backslash |f\backslash |\_1\; =\; \backslash int\; |f|\backslash ,d\backslash mu$ || If the measure ''μ'' on ''S'' is sigma-finite |- ! L∞(μ) | $N\_\backslash mu^\backslash perp$ || || ? || $\backslash |f\backslash |\_\backslash infty\; \backslash equiv\; \backslash inf\; \backslash .$ || where $N\_\backslash mu^\backslash perp\; =\backslash$ |- ! BV(I) | ? || || || $\backslash |f\backslash |\_\; =\; \backslash lim\_f(x)\; +\; V\_f(I)$ || ''V''f(''I'') is the total variation of ''f''. |- ! NBV(I) | ? || || || $\backslash |f\backslash |\_\; =\; V\_f(I)$ || NBV(''I'') consists of BV functions such that $\backslash lim\_f(x)=0$. |- ! AC(I) | K+''L''∞(''I'') || || || $\backslash |f\backslash |\_\; =\; \backslash lim\_f(x)\; +\; V\_f(I)$ || Isomorphic to the Sobolev space ''W''1,1(''I''). |- ! C''n''() || rca(()) || || || $\backslash |f\backslash |\; =\; \backslash sum\_^n\; \backslash sup\_\; |f^(x)|.$ || Isomorphic to R''n'' ⊕ C(()), essentially by Taylor's theorem. |} 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「List of Banach spaces」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.