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

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

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 $A\; \backslash subseteq\; B$ then $B^0\; \backslash subseteq\; A^0$
• For all $\backslash gamma\; \backslash neq\; 0$ : $(\backslash gamma\; A)^0\; =\; \backslash frac\{1\}\{\backslash mid\backslash gamma\backslash mid\}A^0$
• $(\backslash bigcup\_\{i\; \backslash in\; I\}\; A\_i)\; =\; \backslash bigcap\_\{i\; \backslash in\; I\}A\_i^0$
• 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 $A^\{00\}=A$.

Functional analysis