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

Main article: Polar set

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:

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

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

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

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

$${\displaystyle C^{\circ \circ }=\operatorname {cl} C.}$$

Relation to the Fenchel–Moreau theorem

Let

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

be the indicator function for a cone ${\displaystyle C.}$ Then the convex conjugate,

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

is the support function for ${\displaystyle C,}$ 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 − 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

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. 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.