Polar set

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.

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

Properties

${\displaystyle C^{\circ }=\{y\in Y:\sup\{\langle x,y\rangle :x\in C\}\leq 1\}}$.[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 ${\displaystyle x_{0}}$, given by the set of points ${\displaystyle x}$ satisfying ${\displaystyle \langle x,x_{0}\rangle =0}$ is its polar hyperplane, and the dual relationship for a hyperplane yields its pole.

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