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.[1]^:76–77

Preliminaries

Suppose that 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 x ∈ X and ${\displaystyle x^{\prime }\in X^{\prime }}$. 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:

${\displaystyle A^{\circ }:=\left\{x^{\prime }\in X^{\prime }:\sup _{a\in A}\left|\left\langle a,x^{\prime }\right\rangle \right|\leq 1\right\}}$

while the prepolar of a subset B of ${\displaystyle X^{\prime }}$ 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 A of X, often denoted by A^∘∘ is the set

${\displaystyle A^{\circ \circ }:={}^{\circ }\left(A^{\circ }\right)=\left\{x\in X:\sup _{x^{\prime }\in A^{\circ }}\left|\left\langle x,x^{\prime }\right\rangle \right|\leq 1\right\}}$.

Statement in functional analysis

Let ${\displaystyle \sigma \left(X,X^{\prime }\right)}$ denote the weak topology on X (i.e. the weakest TVS topology on X making all linear functionals in ${\displaystyle X^{\prime }}$ continuous).

The bipolar theorem:[2] The bipolar of a subset A of X is equal to the ${\displaystyle \sigma \left(X,X^{\prime }\right)}$-closure of the convex balanced hull of A.

Statement in convex analysis

The bipolar theorem:[1]^:54[3] For any nonempty cone A in some linear space X, the bipolar set A^∘∘ is given by:

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

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.[3][4] 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

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

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

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

is the support function for C, and ${\displaystyle f^{**}(x)=\delta (x|C^{\circ \circ })}$. Therefore, C = C^∘∘ if and only if f = f^**.[1]^:54[4]

See also

• Dual system
• Fenchel–Moreau theorem − A generalization of the bipolar theorem.
• Polar set

References

1. Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701.
2. Narici & Beckenstein 2011, pp. 225-273.
3. Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 9780521833783. Retrieved October 15, 2011.
4. Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–125. ISBN 9780691015866.

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. 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.