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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 Strong topology (polar topology) ： ウィキペディア英語版 Strong topology (polar topology) In functional analysis and related areas of mathematics the strong topology is the finest polar topology, the topology with the most open sets, on a dual pair. The coarsest polar topology is called weak topology. == Definition == Let $(X,Y,\backslash langle\; ,\; \backslash rangle)$ be a dual pair of vector spaces over the field $$ of real ($$) or complex ($$) numbers. Let us denote by $$ the system of all subsets $B\backslash subseteq\; X$ bounded by elements of $Y$ in the following sense: :$$ \forall y\in Y \qquad \sup_|\langle x, y\rangle|<\infty. Then the strong topology $\backslash beta(Y,X)$ on $Y$ is defined as the locally convex topology on $Y$ generated by the seminorms of the form :$$ ||y||_B=\sup_|\langle x, y\rangle|,\qquad y\in Y,\qquad B\in. In the special case when $X$ is a locally convex space, the strong topology on the (continuous) dual space $X\text{'}$ (i.e. on the space of all continuous linear functionals $f:X\backslash to$) is defined as the strong topology $\backslash beta(X\text{'},X)$, and it coincides with the topology of uniform convergence on bounded sets in $X$, i.e. with the topology on $X\text{'}$ generated by the seminorms of the form :$$ ||f||_B=\sup_|f(x)|,\qquad f\in X', where $B$ runs over the family of all bounded sets in $X$. The space $X\text{'}$ with this topology is called strong dual space of the space $X$ and is denoted by $X\text{'}\_\backslash beta$. 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「Strong topology (polar 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.