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.
Preliminaries
Suppose that is a topological vector space (TVS) with a continuous dual space and let for all and The convex hull of a set denoted by is the smallest convex set containing The convex balanced hull of a set is the smallest convex balanced set containing
The polar of a subset is defined to be:
while the prepolar of a subset is: The bipolar of a subset often denoted by is the setStatement in functional analysis
Let denote the weak topology on (that is, the weakest TVS topology on making all linear functionals in continuous).
- The bipolar theorem: The bipolar of a subset is equal to the -closure of the convex balanced hull of
Statement in convex analysis
- The bipolar theorem: For any nonempty cone in some linear space the bipolar set is given by:
Special case
A subset is a nonempty closed convex cone if and only if when where denotes the positive dual cone of a set Or more generally, if 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 Then the convex conjugate, is the support function for and Therefore, if and only ifSee 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)