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 Suppose that

is a topological vector space (TVS) with a continuous dual space

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

while the prepolar of a subset

The bipolar of a subset

often denoted by

σ

denote the weak topology on

(that is, the weakest TVS topology on

making all linear functionals in

= cl ⁡ ( co ⁡ { r a : r ≥ 0 , a ∈

{\displaystyle A^{\circ \circ }=\operatorname {cl} (\operatorname {co} \{ra:r\geq 0,a\in A\}).}

A subset

is a nonempty closed convex cone if and only if

denotes the positive dual cone of a set

is a nonempty convex cone then the bipolar cone is given by

= cl ⁡

{\displaystyle C^{\circ \circ }=\operatorname {cl} C.}

f ( x ) := δ ( x

be the indicator function for a cone

Then the convex conjugate,

is the support function for