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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 dual topology ： ウィキペディア英語版 dual topology In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space. The different dual topologies for a given dual pair are characterized by the Mackey–Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology. Several topological properties depend only on the dual pair and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one. ==Definition== Given a dual pair $(X,\; Y,\; \backslash langle\; ,\; \backslash rangle)$, a dual topology on $X$ is a locally convex topology $\backslash tau$ so that :$(X,\; \backslash tau)\text{'}\; \backslash simeq\; Y.$ Here $(X,\; \backslash tau)\text{'}$ denotes the continuous dual of $(X,\backslash tau)$ and $(X,\; \backslash tau)\text{'}\; \backslash simeq\; Y$ means that there is a linear isomorphism :$\backslash Psi\; :\; Y\; \backslash to\; (X,\; \backslash tau)\text{'},\backslash quad\; y\; \backslash mapsto\; (x\; \backslash mapsto\; \backslash langle\; x,\; y\backslash rangle).$ (If a locally convex topology $\backslash tau$ on $X$ is not a dual topology, then either $\backslash Psi$ is not surjective or it is ill-defined since the linear functional $x\; \backslash mapsto\; \backslash langle\; x,\; y\backslash rangle$ is not continuous on $X$ for some $y$.) 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「dual topology」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.