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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 polar set ： ウィキペディア英語版 polar set :''See also polar set (potential theory).'' In functional analysis and related areas of mathematics the polar set of a given subset of a vector space is a certain set in the dual space. Given a dual pair $(X,Y)$ the polar set or polar of a subset $A$ of $X$ is a set $A^\backslash circ$ in $Y$ defined as :$A^\backslash circ\; :=\; \backslash$ The bipolar of a subset $A$ of $X$ is the polar of $A^\backslash circ$. It is denoted $A^$ and is a set in $X$. == Properties == * $A^\backslash circ$ is absolutely convex * If $A\; \backslash subseteq\; B$ then $B^\backslash circ\; \backslash subseteq\; A^\backslash circ$ * * So $\backslash bigcup\_A\_i^\backslash circ\; \backslash subseteq\; (\backslash bigcap\_\; A\_i)^\backslash circ$, where equality of sets does not necessarily hold. * For all $\backslash gamma\; \backslash neq\; 0$ : $(\backslash gamma\; A)^\backslash circ\; =\; \backslash fracA^\backslash circ$ * $(\backslash bigcup\_\; A\_i)^\backslash circ\; =\; \backslash bigcap\_A\_i^\backslash circ$ * For a dual pair $(X,Y)$ $A^\backslash circ$ is closed in $Y$ under the weak- *-topology on $Y$ * The bipolar $A^$ of a set $A$ is the absolutely convex envelope of $A$, that is the smallest absolutely convex set containing $A$. If $A$ is already absolutely convex then $A^=A$. * For a closed convex cone $C$ in $X$, the polar cone is equivalent to the one-sided polar set for $C$, given by : $C^\backslash circ\; =\; \backslash$. 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「polar set」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.