Bipolar theorem


In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar 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 X is a topological vector space with a continuous dual space and let for all x ∈ X and.
The convex hull of a set A, denoted by co, is the smallest convex set containing A.
The convex balanced hull of a set A is the smallest convex balanced set containing A.
The polar of a subset A of X is defined to be:
while the prepolar of a subset B of is:
The bipolar of a subset A of X, often denoted by A∘∘ is the set

Statement in functional analysis

Let denote the weak topology on X.

Statement in convex analysis

Special case

A subset C of X is a nonempty closed convex cone if and only if C++ = C∘∘ = C when C++ = +, where A+ denotes the positive dual cone of a set A.
Or more generally, if C 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 C.
Then the convex conjugate,
is the support function for C, and.
Therefore, C = C∘∘ if and only if f = f**.