In functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

 and related areas of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 the polar set of a given subset of a vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 is a certain set in the dual space

Dual space

In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...

.

Given a dual pair

Dual pair

In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form....

  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 .

Properties

• is absolutely convex
• If then
• For all :
• 
• For a dual pair is closed
  Closed set
  In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
  
   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 .

Geometry

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, 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.