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 set

Statement 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 if

See also

• − A generalization of the bipolar theorem.

References

Bibliography