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.^1: 76–77

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: $${\displaystyle A^{\circ }:=\left\{x^{\prime }\in X^{\prime }:\sup _{a\in A}\left|\left\langle a,x^{\prime }\right\rangle \right|\leq 1\right\}.}$$ while the prepolar of a subset ${\displaystyle B\subseteq X^{\prime }}$ is: $${\displaystyle {}^{\circ }B:=\left\{x\in X:\sup _{x^{\prime }\in B}\left|\left\langle x,x^{\prime }\right\rangle \right|\leq 1\right\}.}$$ The bipolar of a subset ${\displaystyle A\subseteq X,}$ often denoted by ${\displaystyle A^{\circ \circ }}$ is the set $${\displaystyle A^{\circ \circ }:={}^{\circ }\left(A^{\circ }\right)=\left\{x\in X:\sup _{x^{\prime }\in A^{\circ }}\left|\left\langle x,x^{\prime }\right\rangle \right|\leq 1\right\}.}$$

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

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:

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

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 ${\displaystyle A.}$³⁴ Or more generally, if ${\displaystyle C}$ is a nonempty convex cone then the bipolar cone is given by $${\displaystyle C^{\circ \circ }=\operatorname {cl} C.}$$

Let $${\displaystyle f(x):=\delta (x|C)={\begin{cases}0&x\in C\\\infty &{\text{otherwise}}\end{cases}}}$$ be the indicator function for a cone ${\displaystyle C.}$ Then the convex conjugate, $${\displaystyle f^{*}(x^{*})=\delta \left(x^{*}|C^{\circ }\right)=\delta ^{*}\left(x^{*}|C\right)=\sup _{x\in C}\langle x^{*},x\rangle }$$ is the support function for ${\displaystyle C,}$ and ${\displaystyle f^{**}(x)=\delta (x|C^{\circ \circ }).}$ Therefore, ${\displaystyle C=C^{\circ \circ }}$ if and only if ${\displaystyle f=f^{**}.}$^1: 54 4