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

$A^\circ := \{y \in Y : \sup_{x \in A} |\langle x,y \rangle | \le 1\}$

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

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

1. Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. p. 215. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.

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.