# Bipolar theorem

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In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (i.e. the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem.:76–77

## Preliminaries

Suppose that X is a topological vector space (TVS) with a continuous dual space $X^{\prime }$ and let $\left\langle x,x^{\prime }\right\rangle :=x^{\prime }(x)$ for all xX and $x^{\prime }\in X^{\prime }$ . The convex hull of a set A, denoted by co(A), is the smallest convex set containing A. The convex balanced hull of a set A is the smallest convex balanced set containing A.

The polar of a subset A of X is defined to be:

$A^{\circ }:=\left\{x^{\prime }\in X^{\prime }:\sup _{a\in A}\left|\left\langle a,x^{\prime }\right\rangle \right|\leq 1\right\}$ while the prepolar of a subset B of $X^{\prime }$ is:

${}^{\circ }B:=\left\{x\in X:\sup _{x^{\prime }\in B}\left|\left\langle x,x^{\prime }\right\rangle \right|\leq 1\right\}$ .

The bipolar of a subset A of X, often denoted by A∘∘ is the set

$A^{\circ \circ }:={}^{\circ }\left(A^{\circ }\right)=\left\{x\in X:\sup _{x^{\prime }\in A^{\circ }}\left|\left\langle x,x^{\prime }\right\rangle \right|\leq 1\right\}$ .

## Statement in functional analysis

Let $\sigma \left(X,X^{\prime }\right)$ denote the weak topology on X (i.e. the weakest TVS topology on X making all linear functionals in $X^{\prime }$ continuous).

The bipolar theorem: The bipolar of a subset A of X is equal to the $\sigma \left(X,X^{\prime }\right)$ -closure of the convex balanced hull of A.

## Statement in convex analysis

The bipolar theorem::54 For any nonempty cone A in some linear space X, the bipolar set A∘∘ is given by:
$A^{\circ \circ }=\operatorname {cl} \left(\operatorname {co} \left\{ra:r\geq 0,a\in A\right\}\right)$ .

### Special case

A subset C of X is a nonempty closed convex cone if and only if C++ = C∘∘ = C when C++ = (C+)+, where A+ denotes the positive dual cone of a set A. Or more generally, if C is a nonempty convex cone then the bipolar cone is given by

C∘∘ = cl(C).

## Relation to the Fenchel–Moreau theorem

Let

$f(x):=\delta \left(x|C\right)={\begin{cases}0&x\in C\\\infty &{\text{otherwise}}\end{cases}}$ be the indicator function for a cone C. Then the convex conjugate,

$f^{*}(x^{*})=\delta (x^{*}|C^{\circ })=\delta ^{*}(x^{*}|C)=\sup _{x\in C}\langle x^{*},x\rangle$ is the support function for C, and $f^{**}(x)=\delta (x|C^{\circ \circ })$ . Therefore, C = C∘∘ if and only if f = f**.:54