Polar set

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

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

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

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 $A^{00}=A$.