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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 Uniformly smooth space ： ウィキペディア英語版 Uniformly smooth space In mathematics, a uniformly smooth space is a normed vector space $X$ satisfying the property that for every $\backslash epsilon>0$ there exists $\backslash delta>0$ such that if $x,y\backslash in\; X$ with $\backslash |x\backslash |=1$ and $\backslash |y\backslash |\backslash leq\backslash delta$ then :$\backslash |x+y\backslash |+\backslash |x-y\backslash |\; \backslash le\; 2\; +\; \backslash epsilon\backslash |y\backslash |.$ The modulus of smoothness of a normed space ''X'' is the function ρ''X'' defined for every by the formula〔see Definition 1.e.1, p. 59 in .〕 :$\backslash rho\_X(t)\; =\; \backslash sup\; \backslash Bigl\backslash \; -\; 1\; \backslash ,:\backslash ,\; \backslash |x\backslash |\; =\; 1,\; \backslash ;\; \backslash |y\backslash |\; =\; t\; \backslash Bigr\backslash \}.$ The triangle inequality yields that . The normed space ''X'' is uniformly smooth if and only if tends to 0 as ''t'' tends to 0. ==Properties== * Every uniformly smooth Banach space is reflexive.〔Proposition 1.e.3, p. 61 in .〕 * A Banach space $X$ is uniformly smooth if and only if its continuous dual $X^$ * is uniformly convex (and vice versa, via reflexivity).〔Proposition 1.e.2, p. 61 in .〕 The moduli of convexity and smoothness are linked by ::$\backslash rho\_(t)\; =\; \backslash sup\; \backslash ,\; \backslash quad\; t\; \backslash ge\; 0,$ :and the maximal convex function majorated by the modulus of convexity δ''X'' is given by〔Proposition 1.e.6, p. 65 in .〕 ::$\backslash tilde\; \backslash delta\_X(\backslash varepsilon)\; =\; \backslash sup\; \backslash .$ :Furthermore,〔Lemma 1.e.7 and 1.e.8, p. 66 in .〕 ::$\backslash delta\_X(\backslash varepsilon\; /\; 2)\; \backslash le\; \backslash tilde\; \backslash delta\_X(\backslash varepsilon)\; \backslash le\; \backslash delta\_X(\backslash varepsilon),\; \backslash quad\; \backslash varepsilon\; \backslash in\; (2).$ * A Banach space is uniformly smooth if and only if the limit ::$\backslash lim\_\backslash frac$ :exists uniformly for all $x,\; y\backslash in\; S\_X$ (where $S\_X$ denotes the unit sphere of $X$). *When , the ''L''''p''-spaces are uniformly smooth (and uniformly convex). Enflo proved〔Enflo, Per (1973), "Banach spaces which can be given an equivalent uniformly convex norm", Israel J. Math. 13:281–288.〕 that the class of Banach spaces that admit an equivalent uniformly convex norm coincides with the class of super-reflexive Banach spaces, introduced by Robert C. James.〔James, Robert C. (1972), "Super-reflexive Banach spaces", Canad. J. Math. 24:896–904.〕 As a space is super-reflexive if and only if its dual is super-reflexive, it follows that the class of Banach spaces that admit an equivalent uniformly convex norm coincides with the class of spaces that admit an equivalent uniformly smooth norm. The Pisier renorming theorem〔Pisier, Gilles (1975), "Martingales with values in uniformly convex spaces", Israel J. Math. 20:326–350.〕 states that a super-reflexive space ''X'' admits an equivalent uniformly smooth norm for which the modulus of smoothness ρ''X'' satisfies, for some constant ''C'' and some  :$\backslash rho\_X(t)\; \backslash le\; C\; \backslash ,\; t^p,\; \backslash quad\; t\; >\; 0.$ It follows that every super-reflexive space ''Y'' admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant  and some positive real ''q'' :$\backslash delta\_Y(\backslash varepsilon)\; \backslash ge\; c\; \backslash ,\; \backslash varepsilon^q,\; \backslash quad\; \backslash varepsilon\; \backslash in\; (2).$ If a normed space admits two equivalent norms, one uniformly convex and one uniformly smooth, the Asplund averaging technique〔Asplund, Edgar (1967), "Averaged norms", Israel J. Math. 5:227–233.〕 produces another equivalent norm that is both uniformly convex and uniformly smooth. 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「Uniformly smooth space」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.