Polar set: Difference between revisions
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:''See also [[polar set (potential theory)]].'' |
:''See also [[polar set (potential theory)]].'' |
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In [[functional analysis]] and related areas of [[mathematics]] |
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]]. |
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Given a [[dual pair]] <math>(X,Y)</math> the '''polar''' of a subset <math>A</math> of <math>X</math> is a set <math>A^0</math> in <math>Y</math> defined as |
Given a [[dual pair]] <math>(X,Y)</math> the '''polar set''' or '''polar''' of a subset <math>A</math> of <math>X</math> is a set <math>A^0</math> in <math>Y</math> defined as |
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:<math>A^0 := \{y \in Y : \sup\{\mid \langle x,y \rangle \mid : x \in A \} \le 1\}</math> |
:<math>A^0 := \{y \in Y : \sup\{\mid \langle x,y \rangle \mid : x \in A \} \le 1\}</math> |
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Revision as of 16:56, 8 May 2007
- 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 the polar set or polar of a subset of is a set in defined as
The bipolar of a subset of is the polar of . It is denoted and is a set in .
Properties
- is absolutely convex
- If then
- For all :
- For a dual pair is closed in under the weak-*-topology on
- The bipolar of a set is the absolutely convex envelope of , that is the smallest absolutely convex set containing . If is already absolutely convex then .