Bipolar theorem
# Bipolar theorem

In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, 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 $x\in X$ and $x^{\prime }\in X^{\prime }.$ The convex hull of a set $A,$ denoted by $\operatorname {co} A,$ 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 $A\subseteq X$ 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 $B\subseteq X^{\prime }$ 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 $A\subseteq X,$ often denoted by $A^{\circ \circ }$ 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 $X$ (that is, the weakest TVS topology on $A$ making all linear functionals in $X^{\prime }$ continuous).

    The bipolar theorem: The bipolar of a subset $A\subseteq X$ 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 $A^{\circ \circ }$ is given by:

$A^{\circ \circ }=\operatorname {cl} (\operatorname {co} \{ra:r\geq 0,a\in A\}).$

### Special case

A subset $C\subseteq X$ is a nonempty closed convex cone if and only if $C^{++}=C^{\circ \circ }=C$ when $C^{++}=\left(C^{+}\right)^{+},$ 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

$C^{\circ \circ }=\operatorname {cl} C.$

## Relation to the Fenchel–Moreau theorem

Let

$f(x):=\delta (x|C)={\begin{cases}0&x\in C\\\infty &{\text{otherwise}}\end{cases}}$

be the indicator function for a cone $C.$ Then the convex conjugate,

$f^{*}(x^{*})=\delta \left(x^{*}|C^{\circ }\right)=\delta ^{*}\left(x^{*}|C\right)=\sup _{x\in C}\langle x^{*},x\rangle$

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

