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 ${\displaystyle \left\langle x,x^{\prime }\right\rangle$ :=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 $X$ making all linear functionals in $X^{\prime }$ continuous).

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

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

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 $f=f^{**}.$

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)

Bibliography

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.