Bipolar theorem

In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (i.e. 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 $\left\langle x,x^{\prime }\right\rangle :=x^{\prime }(x)$ for all x ∈ X and $x^{\prime }\in X^{\prime }$. The convex hull of a set A, denoted by 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 of 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 of $X^{\prime }$ 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 of X, often denoted by A^∘∘ 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 (i.e. the weakest TVS topology on X making all linear functionals in $X^{\prime }$ continuous).

The bipolar theorem: The bipolar of a subset A of 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^∘∘ is given by:

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

Special case

A subset C of X is a nonempty closed convex cone if and only if C⁺⁺ = C^∘∘ = C when C⁺⁺ = (C⁺)⁺, 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^∘∘ = cl(C).

Relation to the Fenchel–Moreau theorem

Let

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

be the indicator function for a cone C. Then the convex conjugate,

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

is the support function for C, and $f^{**}(x)=\delta (x|C^{\circ \circ })$. Therefore, C = C^∘∘ if and only if f = f^**.

See also

• Dual system
• Polar set

References

• 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.