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.

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: For any nonempty cone $$A$$ in some linear space $$X,$$ the bipolar set $$A^{\circ \circ}$$ is given by:

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

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