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.^{[1]: 76–77}

Preliminaries

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

The polar of a subset ${\displaystyle A\subseteq X}$ is defined to be:

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

${\displaystyle B\subseteq X^{\prime ))$

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

${\displaystyle A\subseteq X,}$

${\displaystyle A^{\circ \circ ))$

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

Statement in functional analysis

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

The bipolar theorem:^[2] The bipolar of a subset ${\displaystyle A\subseteq X}$ is equal to the ${\displaystyle \sigma \left(X,X^{\prime }\right)}$-closure of the convex balanced hull of ${\displaystyle A.}$

Statement in convex analysis

The bipolar theorem:^{[1]: 54 [3]} For any nonempty cone ${\displaystyle A}$ in some linear space ${\displaystyle X,}$ the bipolar set ${\displaystyle A^{\circ \circ ))$ is given by:

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

Special case

A subset ${\displaystyle C\subseteq X}$ is a nonempty closed convex cone if and only if ${\displaystyle C^{++}=C^{\circ \circ }=C}$ when ${\displaystyle C^{++}=\left(C^{+}\right)^{+},}$ where ${\displaystyle A^{+))$ denotes the positive dual cone of a set ${\displaystyle A.}$^[3][4] Or more generally, if ${\displaystyle C}$ is a nonempty convex cone then the bipolar cone is given by

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

Relation to the Fenchel–Moreau theorem

Let

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

${\displaystyle C.}$

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

${\displaystyle C,}$

${\displaystyle f^{**}(x)=\delta (x|C^{\circ \circ }).}$

${\displaystyle C=C^{\circ \circ ))$

${\displaystyle f=f^{**}.}$