Polar set

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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 the polar set or polar of a subset of is a set in defined as

The bipolar of a subset of is the polar of . It is denoted and is a set in .


  • is absolutely convex
  • If then
    • So , where equality of sets does not necessarily hold.
  • For all  :
  • For a dual pair is closed in under the weak-*-topology on
  • The bipolar of a set is the absolutely convex envelope of , that is the smallest absolutely convex set containing . If is already absolutely convex then .
  • For a closed convex cone in , the polar cone is equivalent to the one-sided polar set for , given by
. [1]


In geometry, the polar set may also refer to a duality between points and planes. In particular, the polar set of a point , given by the set of points satisfying is its polar hyperplane, and the dual relationship for a hyperplane yields its pole. In convex geometry the polar set of a convex set containing the origin is defined similarly but without taking the absolute values of the inner product in the definition, resulting in a different set.

See also[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. 

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.