# 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 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 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

## Geometry

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.*

## See also

## References

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- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}

**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.