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

Main article: Polar set

Suppose that $X$ is a topological vector space (TVS) with a continuous dual space $X^{\prime }$ and let $⟨x,x^{\prime }⟩:=x^{\prime }(x)$ for all $x\in X$ and $x^{\prime }\in X^{\prime }.$ The convex hull of a set $A,$ denoted by $\mathrm{co}A,$ 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: $$A^{\circ }:=\{x^{\prime }\in X^{\prime }:\underset{a\in A}{\sup }\left|⟨a,x^{\prime }⟩\right|\le 1\}.$$ while the prepolar of a subset $B\subseteq X^{\prime }$ is: $${}^{\circ }B:=\{x\in X:\underset{x^{\prime }\in B}{\sup }\left|⟨x,x^{\prime }⟩\right|\le 1\}.$$ The bipolar of a subset $A\subseteq X,$ often denoted by $A^{\circ \circ }$ is the set $$A^{\circ \circ }:={}^{\circ }\left(A^{\circ }\right)=\{x\in X:\underset{x^{\prime }\in A^{\circ }}{\sup }\left|⟨x,x^{\prime }⟩\right|\le 1\}.$$

Statement in functional analysis

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

The bipolar theorem: The bipolar of a subset $A\subseteq X$ is equal to the $\sigma (X,X^{\prime })$-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:

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

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

Relation to the Fenchel–Moreau theorem

Let $$f(x):=\delta (x|C)=\{\begin{array}{cc}0& x\in C\\ \mathrm{\infty }& \text{otherwise}\end{array}$$ be the indicator function for a cone $C.$ Then the convex conjugate, $$f^{*}(x^{*})=\delta (x^{*}|C^{\circ })={\delta }^{*}(x^{*}|C)=\underset{x\in C}{\sup }⟨x^{*},x⟩$$ is the support function for $C,$ and $f^{**}(x)=\delta (x|C^{\circ \circ }).$ Therefore, $C=C^{\circ \circ }$ if and only if $f=f^{**}.$

See also

• Dual system – Dual pair of vector spaces
• Fenchel–Moreau theorem – Mathematical theorem in convex analysis − A generalization of the bipolar theorem.
• Polar set – Subset of all points that is bounded by some given point of a dual (in a dual pairing)