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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 Polynomially reflexive space ： ウィキペディア英語版 Polynomially reflexive space In mathematics, a polynomially reflexive space is a Banach space ''X'', on which the space of all polynomials in each degree is a reflexive space. Given a multilinear functional ''M''''n'' of degree ''n'' (that is, ''M''''n'' is ''n''-linear), we can define a polynomial ''p'' as :$p(x)=M\_n(x,\backslash dots,x)$ (that is, applying ''M''''n'' on the ''diagonal'') or any finite sum of these. If only ''n''-linear functionals are in the sum, the polynomial is said to be ''n''-homogeneous. We define the space ''P''''n'' as consisting of all ''n''-homogeneous polynomials. The ''P''1 is identical to the dual space, and is thus reflexive for all reflexive ''X''. This implies that reflexivity is a prerequisite for polynomial reflexivity. ==Relation to continuity of forms== On a finite-dimensional linear space, a quadratic form ''x''↦''f''(''x'') is always a (finite) linear combination of products ''x''↦''g''(''x'') ''h''(''x'') of two linear functionals ''g'' and ''h''. Therefore, assuming that the scalars are complex numbers, every sequence ''xn'' satisfying ''g''(''xn'') → 0 for all linear functionals ''g'', satisfies also ''f''(''xn'') → 0 for all quadratic forms ''f''. In infinite dimension the situation is different. For example, in a Hilbert space, an orthonormal sequence ''xn'' satisfies ''g''(''xn'') → 0 for all linear functionals ''g'', and nevertheless ''f''(''xn'') = 1 where ''f'' is the quadratic form ''f''(''x'') = ||''x''||2. In more technical words, this quadratic form fails to be weakly sequentially continuous at the origin. On a reflexive Banach space with the approximation property the following two conditions are equivalent:〔Farmer 1994, page 261.〕 * every quadratic form is weakly sequentially continuous at the origin; * the Banach space of all quadratic forms is reflexive. Quadratic forms are 2-homogeneous polynomials. The equivalence mentioned above holds also for ''n''-homogeneous polynomials, ''n''=3,4,... 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「Polynomially reflexive 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.