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 $A^{\circ }$ in $Y$ defined as

$A^{\circ }:=\{y\in Y:\sup _{x\in A}|\langle x,y\rangle |\leq 1\}$

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

Properties

• $A^{\circ }$ is absolutely convex
• If $A\subseteq B$ then $B^{\circ }\subseteq A^{\circ }$
  • So $\bigcup _{i\in I}A_{i}^{\circ }\subseteq (\bigcap _{i\in I}A_{i})^{\circ }$, where equality of sets does not necessarily hold.
• For all $\gamma \neq 0$ : $(\gamma A)^{\circ }={\frac {1}{\mid \gamma \mid }}A^{\circ }$
• $(\bigcup _{i\in I}A_{i})^{\circ }=\bigcap _{i\in I}A_{i}^{\circ }$
• For a dual pair $(X,Y)$ $A^{\circ }$ is closed in $Y$ under the weak-*-topology on $Y$
• The bipolar $A^{\circ \circ }$ 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 $A^{\circ \circ }=A$.
• For a closed convex cone $C$ in $X$, the polar cone is equivalent to the one-sided polar set for $C$, given by

$C^{\circ }=\{y\in Y:\sup\{\langle x,y\rangle :x\in C\}\leq 1\}$.

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 $\langle x,x_{0}\rangle =0$ is its polar hyperplane, and the dual relationship for a hyperplane yields its pole.

See also

• Polar cone
• Bipolar theorem

References

Discussion of Polar Sets in Potential Theory: Ransford, Thomas: Potential Theory in the Complex Plane, London Mathematical Society Student Texts 28, CUP, 1995, pp. 55-58.