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 [math]\displaystyle{ X }[/math] is a topological vector space (TVS) with a continuous dual space [math]\displaystyle{ X^{\prime} }[/math] and let [math]\displaystyle{ \left\langle x, x^{\prime} \right\rangle := x^{\prime}(x) }[/math] for all [math]\displaystyle{ x \in X }[/math] and [math]\displaystyle{ x^{\prime} \in X^{\prime}. }[/math] The convex hull of a set [math]\displaystyle{ A, }[/math] denoted by [math]\displaystyle{ \operatorname{co} A, }[/math] is the smallest convex set containing [math]\displaystyle{ A. }[/math] The convex balanced hull of a set [math]\displaystyle{ A }[/math] is the smallest convex balanced set containing [math]\displaystyle{ A. }[/math]
The polar of a subset [math]\displaystyle{ A \subseteq X }[/math] is defined to be: [math]\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\}. }[/math] while the prepolar of a subset [math]\displaystyle{ B \subseteq X^{\prime} }[/math] is: [math]\displaystyle{ {}^{\circ} B := \left\{ x \in X : \sup_{x^{\prime} \in B} \left| \left\langle x, x^{\prime} \right\rangle \right| \leq 1 \right\}. }[/math] The bipolar of a subset [math]\displaystyle{ A \subseteq X, }[/math] often denoted by [math]\displaystyle{ A^{\circ\circ} }[/math] is the set [math]\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\}. }[/math]
Statement in functional analysis
Let [math]\displaystyle{ \sigma\left(X, X^{\prime}\right) }[/math] denote the weak topology on [math]\displaystyle{ X }[/math] (that is, the weakest TVS topology on [math]\displaystyle{ A }[/math] making all linear functionals in [math]\displaystyle{ X^{\prime} }[/math] continuous).
- The bipolar theorem:[2] The bipolar of a subset [math]\displaystyle{ A \subseteq X }[/math] is equal to the [math]\displaystyle{ \sigma\left(X, X^{\prime}\right) }[/math]-closure of the convex balanced hull of [math]\displaystyle{ A. }[/math]
Statement in convex analysis
- The bipolar theorem:[1]:54[3] For any nonempty cone [math]\displaystyle{ A }[/math] in some linear space [math]\displaystyle{ X, }[/math] the bipolar set [math]\displaystyle{ A^{\circ \circ} }[/math] is given by:
[math]\displaystyle{ A^{\circ \circ} = \operatorname{cl} (\operatorname{co} \{ r a : r \geq 0, a \in A \}). }[/math]
Special case
A subset [math]\displaystyle{ C \subseteq X }[/math] is a nonempty closed convex cone if and only if [math]\displaystyle{ C^{++} = C^{\circ \circ} = C }[/math] when [math]\displaystyle{ C^{++} = \left(C^{+}\right)^{+}, }[/math] where [math]\displaystyle{ A^{+} }[/math] denotes the positive dual cone of a set [math]\displaystyle{ A. }[/math][3][4] Or more generally, if [math]\displaystyle{ C }[/math] is a nonempty convex cone then the bipolar cone is given by [math]\displaystyle{ C^{\circ \circ} = \operatorname{cl} C. }[/math]
Relation to the Fenchel–Moreau theorem
Let [math]\displaystyle{ f(x) := \delta(x|C) = \begin{cases}0 & x \in C\\ \infty & \text{otherwise}\end{cases} }[/math] be the indicator function for a cone [math]\displaystyle{ C. }[/math] Then the convex conjugate, [math]\displaystyle{ f^*(x^*) = \delta\left(x^*|C^\circ\right) = \delta^*\left(x^*|C\right) = \sup_{x \in C} \langle x^*,x \rangle }[/math] is the support function for [math]\displaystyle{ C, }[/math] and [math]\displaystyle{ f^{**}(x) = \delta(x|C^{\circ\circ}). }[/math] Therefore, [math]\displaystyle{ C = C^{\circ \circ} }[/math] if and only if [math]\displaystyle{ f = f^{**}. }[/math][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
- ↑ 1.0 1.1 1.2 Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701.
- ↑ Narici & Beckenstein 2011, pp. 225-273.
- ↑ 3.0 3.1 Boyd, Stephen P.; Vandenberghe, Lieven (2004) (pdf). Convex Optimization. Cambridge University Press. pp. 51–53. ISBN 9780521833783. https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=65. Retrieved October 15, 2011.
- ↑ 4.0 4.1 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 (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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