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.^{[1]: 76–77}

Preliminaries

Suppose that ${\displaystyle X}$ 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 ${\displaystyle x\in X}$ and ${\displaystyle x^{\prime }\in X^{\prime }.}$ The convex hull of a set ${\displaystyle A,}$ denoted by ${\displaystyle \operatorname {co} A,}$ is the smallest convex set containing ${\displaystyle A.}$ The convex balanced hull of a set ${\displaystyle A}$ is the smallest convex balanced set containing ${\displaystyle A.}$

The polar of a subset ${\displaystyle A\subseteq X}$ is defined to be:

while the prepolar of a subset ${\displaystyle B\subseteq X^{\prime }}$ is: The bipolar of a subset ${\displaystyle A\subseteq X,}$ often denoted by ${\displaystyle A^{\circ \circ }}$ is the set

Statement in functional analysis

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

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

Statement in convex analysis

The bipolar theorem:^{[1]: 54 [3]} For any nonempty cone ${\displaystyle A}$ in some linear space ${\displaystyle X,}$ the bipolar set ${\displaystyle A^{\circ \circ }}$ is given by:

Special case

A subset ${\displaystyle C\subseteq X}$ is a nonempty closed convex cone if and only if ${\displaystyle C^{++}=C^{\circ \circ }=C}$ when ${\displaystyle C^{++}=\left(C^{+}\right)^{+},}$ where ${\displaystyle A^{+}}$ denotes the positive dual cone of a set ${\displaystyle A.}$^[3][4] Or more generally, if ${\displaystyle C}$ is a nonempty convex cone then the bipolar cone is given by

Relation to the Fenchel–Moreau theorem

Let

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

See also

• Dual system
• 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)

References

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.