View source for Bipolar theorem

{{Short description|Theorem in convex analysis}}
In [[Mathematics|mathematics]], the '''bipolar theorem''' is a [[Theorem|theorem]] in [[Functional analysis|functional analysis]] that characterizes the bipolar (that is, the [[Polar set|polar]] of the polar) of a set. 
In [[Convex analysis|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]].<ref name="BorweinLewis">{{cite book |last1=Borwein |first1=Jonathan |last2=Lewis |first2=Adrian |title=Convex Analysis and Nonlinear Optimization: Theory and Examples| edition=2 |year=2006 |publisher=Springer |isbn=9780387295701}}</ref>{{rp|76–77}}

==Preliminaries==
{{Main|Polar set}}

Suppose that <math>X</math> is a [[Topological vector space|topological vector space]] (TVS) with a continuous dual space <math>X^{\prime}</math> and let <math>\left\langle x, x^{\prime} \right\rangle := x^{\prime}(x)</math> for all <math>x \in X</math> and <math>x^{\prime} \in X^{\prime}.</math> 
The '''[[Convex hull|convex hull]]''' of a set <math>A,</math> denoted by <math>\operatorname{co} A,</math> is the smallest [[Convex set|convex set]] containing <math>A.</math> 
The '''convex balanced hull''' of a set <math>A</math> is the smallest [[Convex set|convex]] [[Balanced set|balanced]] set containing <math>A.</math>

The '''[[Polar set|polar]]''' of a subset <math>A \subseteq X</math> is defined to be: 
<math display="block">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>B \subseteq X^{\prime}</math> is:
<math display="block">{}^{\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>A \subseteq X,</math> often denoted by <math>A^{\circ\circ}</math> is the set 
<math display="block">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>\sigma\left(X, X^{\prime}\right)</math> denote the [[Weak topology|weak topology]] on <math>X</math> (that is, the weakest TVS topology on <math>A</math> making all linear functionals in <math>X^{\prime}</math> continuous).

:'''The bipolar theorem''':{{sfn|Narici|Beckenstein| 2011|pp=225-273}} The bipolar of a subset <math>A \subseteq X</math> is equal to the <math>\sigma\left(X, X^{\prime}\right)</math>-closure of the convex balanced hull of <math>A.</math>

==Statement in convex analysis==

:'''The bipolar theorem''':<ref name="BorweinLewis"/>{{rp|54}}<ref name="Boyd">{{cite book|title=Convex Optimization|first1=Stephen P.|last1=Boyd|first2=Lieven|last2=Vandenberghe|year=2004|publisher=Cambridge University Press|isbn=9780521833783|url=https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=65|format=pdf|accessdate=October 15, 2011|pages=51–53}}</ref> For any nonempty cone <math>A</math> in some linear space <math>X,</math> the bipolar set <math>A^{\circ \circ}</math> is given by: 
<math display="block">A^{\circ \circ} = \operatorname{cl} (\operatorname{co} \{ r a : r \geq 0, a \in A \}).</math>

===Special case===

A subset <math>C \subseteq X</math> is a nonempty [[Closure (mathematics)|closed]] [[Convex cone|convex cone]] if and only if <math>C^{++} = C^{\circ \circ} = C</math> when <math>C^{++} = \left(C^{+}\right)^{+},</math> where <math>A^{+}</math> denotes the positive dual cone of a set <math>A.</math><ref name="Boyd"/><ref name="Rockafellar">{{cite book|author=Rockafellar, R. Tyrrell|title=Convex Analysis|publisher=Princeton University Press|location=Princeton, NJ|year=1997|origyear=1970|isbn=9780691015866|pages=121–125}}</ref>
Or more generally, if <math>C</math> is a nonempty convex cone then the bipolar cone is given by 
<math display="block">C^{\circ \circ} = \operatorname{cl} C.</math>

==Relation to the [[Fenchel–Moreau theorem]]==

Let 
<math display="block">f(x) := \delta(x|C) = \begin{cases}0 & x \in C\\ \infty & \text{otherwise}\end{cases}</math> 
be the [[Characteristic function (convex analysis)|indicator function]] for a cone <math>C.</math>
Then the [[Convex conjugate|convex conjugate]], 
<math display="block">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|support function]] for <math>C,</math> and <math>f^{**}(x) = \delta(x|C^{\circ\circ}).</math> 
Therefore, <math>C = C^{\circ \circ}</math> if and only if <math>f = f^{**}.</math><ref name="BorweinLewis"/>{{rp|54}}<ref name="Rockafellar"/>

==See also==

* {{annotated link|Dual system}}
* {{annotated link|Fenchel–Moreau theorem}} − A generalization of the bipolar theorem.
* {{annotated link|Polar set}}

==References==

{{reflist}}

==Bibliography==

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|2011|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|1999|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->


[[Category:Convex analysis]]
[[Category:Functional analysis]]
[[Category:Theorems in analysis]]

{{Sourceattribution|Bipolar theorem}}