- 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 <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
- <math>A^0 := \{y \in Y : \sup\{\mid \langle x,y \rangle \mid : x \in A \} \le 1\}</math>
The bipolar of a subset <math>A</math> of <math>X</math> is the polar of <math>A^0</math>. It is denoted <math>A^{00}</math> and is a set in <math>X</math>.
Properties
- <math>A^0</math> is absolutely convex
- If <math>A \subseteq B</math> then <math>B^0 \subseteq A^0</math>
- For all <math>\gamma \neq 0</math> : <math>(\gamma A)^0 = \frac{1}{\mid\gamma\mid}A^0</math>
- <math>(\bigcup_{i \in I} A_i)^0 = \bigcap_{i \in I}A_i^0</math>
- For a dual pair <math>(X,Y)</math> <math>A^0</math> is closed in <math>Y</math> under the weak-*-topology on <math>Y</math>
- The bipolar <math>A^{00}</math> of a set <math>A</math> is the absolutely convex envelope of <math>A</math>, that is the smallest absolutely convex set containing <math>A</math>. If <math>A</math> is already absolutely convex then <math>A^{00}=A</math>.
