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In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a dual pair. == Definitions == Let be a dual pair of vector spaces and over the field , either the real or complex numbers. A set is said to be ''bounded'' in with respect to , if for each element the set of values is bounded: : This condition is equivalent to the requirement that the polar of the set in : is an absorbent set in , i.e. : be a family of bounded sets in (with respect to ) with the following properties: * each point of belongs to some set : * each two sets and are contained in some set : : * is closed under the operation of multiplication by scalars: : Then the seminorms of the form : define a Hausdorff locally convex topology on which is called the polar topology on generated by the family of sets . The sets : form a local base of this topology. A net of elements tends to an element in this topology if and only if : Because of this the polar topology is often called the topology of uniform convergence on the sets of . The semi norm is the gauge of the polar set . 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
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