Polar set

:"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 $(X,Y)$ the polar set or polar of a subset $A$ of $X$ is a set $A^0$ in $Y$ defined as:$A^0\; :=\; \{y\; in\; Y\; :\; sup\{mid\; langle\; x,y\; angle\; mid\; :\; x\; in\; A\; \}\; le\; 1\}$

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

Properties

* $A^0$ is absolutely convex
* If $A\; subseteq\; B$ then $B^0\; subseteq\; A^0$
* For all $gamma\; eq\; 0$ : $(gamma\; A)^0\; =\; frac\{1\}\{midgammamid\}A^0$
*
* 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 $A^\{00\}=A$.

Geometry

In geometry, the polar set may also refer to a duality between points and planes. In particular, the polar set of a point $x\_0$, given by the set of points $x$ satisfying $langle\; x,\; x\_0\; angle=0$ is its "polar hyperplane," and the dual relationship for a hyperplane yields its "pole."