Polar set

See also polar set (potential theory).

In functional analysis and related areas of mathematics a polar set of a subset of a vector space is a set in the dual space.

Given a dual pair $(X, Y)$ the polar of a subset $A$ of $X$ is a set $A 0$ in $Y$ defined as

$A^0 := \{y \in Y : \sup\{\mid \langle x,y \rangle \mid : x \in A \} \le 1\}$

The bipolar of a subset $A$ of $X$ is the polar of $A 0$ . It is denoted $A 00$ and is a set in $X$ .

[edit] Properties

$A 0$ is absolutely convex
If $A \subseteq B$ then $B^0 \subseteq A^0$
For all $\gamma \neq 0$ : $(\gamma A)^0 = \frac{1}{\mid\gamma\mid}A^0$
$(\bigcup_{i \in I} A_i)^0 = \bigcap_{i \in I}A_i^0$
For a dual pair $(X, Y)$ $A 0$ is closed in $Y$ under the weak-*-topology on $Y$
The bipolar $A 00$ 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 $A 00 = A$ .

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