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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 topology ： ウィキペディア英語版 polar topology In functional analysis and related areas of mathematics a polar topology, topology of $\backslash mathcal$-convergence or topology of uniform convergence on the sets of $\backslash mathcal$ is a method to define locally convex topologies on the vector spaces of a dual pair. == Definitions == Let $(X,Y,\backslash langle\; ,\; \backslash rangle)$ be a dual pair of vector spaces $X$ and $Y$ over the field $\backslash mathbb$, either the real or complex numbers. A set $A\backslash subseteq\; X$ is said to be ''bounded'' in $X$ with respect to $Y$, if for each element $y\backslash in\; Y$ the set of values $\backslash$ is bounded: :$$ \forall y\in Y\qquad \sup_|\langle x,y\rangle|<\infty. This condition is equivalent to the requirement that the polar $A^\backslash circ$ of the set $A$ in $Y$ :$$ A^\circ=\ is an absorbent set in $Y$, i.e. :$$ \bigcup_ be a family of bounded sets in $X$ (with respect to $Y$) with the following properties: * each point $x$ of $X$ belongs to some set $A\backslash in$ : $$ \forall x\in X\qquad \exists A\in \qquad x\in A, * each two sets $A\backslash in$ and $B\backslash in$ are contained in some set $C\backslash in$: : $$ \forall A,B\in \qquad \exists C\in \qquad A\cup B\subseteq C, * $$ is closed under the operation of multiplication by scalars: : $$ \forall A\in \qquad \forall\lambda\in\qquad \lambda\cdot A\in . Then the seminorms of the form :$$ \|y\|_A=\sup_|\langle x,y\rangle|,\qquad A\in, define a Hausdorff locally convex topology on $Y$ which is called the polar topology on $Y$ generated by the family of sets $$. The sets : $$ U_=\,\qquad B\in , form a local base of this topology. A net of elements $y\_i\backslash in\; Y$ tends to an element $y\backslash in\; Y$ in this topology if and only if : $$ \forall A\in\qquad \|y_i-y\|_A = \sup_ |\langle x,y_i\rangle-\langle x,y\rangle|\underset0. Because of this the polar topology is often called the topology of uniform convergence on the sets of $\backslash mathcal$. The semi norm $\backslash |y\backslash |\_A$ is the gauge of the polar set $A^\backslash circ$. 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 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.