Polar set

From Wikipedia, the free encyclopedia
Jump to: navigation, search
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[edit]

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

Geometry[edit]

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[edit]

References[edit]

  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.