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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 Goldstine theorem ： ウィキペディア英語版 Goldstine theorem In functional analysis, a branch of mathematics, the Goldstine theorem, named after Herman Goldstine, is stated as follows: :Goldstine Theorem. Let be a Banach space, then the image of the closed unit ball under the canonical embedding into the closed unit ball of the bidual space is weak *-dense. The conclusion of the theorem is not true for the norm topology, which can be seen by considering the Banach space of real sequences that converge to zero, , and its bi-dual space . == Proof == Given , an -tuple of linearly independent elements of and a we shall find in such that for . If the requirement is dropped, the existence of such an follows from the surjectivity of :$\backslash begin\; \backslash Phi\; :\; X\; \backslash to\; \backslash mathbf^,\; \backslash \backslash \; x\; \backslash mapsto\; \backslash left\; (\backslash varphi\_1(x),\; \backslash cdots,\; \backslash varphi\_n(x)\; \backslash right\; )\; \backslash end$ Now let :$Y\; :=\; \backslash bigcap\_i\; \backslash ker\; \backslash varphi\_i\; =\; \backslash ker\; \backslash Phi.$ Every element of has the required property, so that it suffices to show that the latter set is not empty. Assume that it is empty. Then and by the Hahn-Banach theorem there exists a linear form such that and . Then and therefore :$1+\backslash delta\; \backslash leq\; \backslash varphi(x)\; =\; x\text{'}\text{'}(\backslash varphi)\; \backslash leq\; \backslash |\backslash varphi\backslash |\_\; \backslash left\; \backslash |x\text{'}\text{'}\; \backslash right\; \backslash |\_\; \backslash leq\; 1,$ which is a contradiction. 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「Goldstine theorem」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.