mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the bipolar theorem is a

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...

that characterizes the bipolar (that is, the polar of the polar) of a set. In

convex analysis Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex optimization, convex minimization, a subdomain of optimization (mathematics), optimization theor ...

, the bipolar theorem refers to a

necessary and sufficient conditions In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...

for a

cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...

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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

(TVS) with a continuous dual space

X^

and let

\left\langle x, x^ \right\rangle := x^(x)

for all

x \in X

and

x^ \in X^.

The

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

of a set

A,

denoted by

\operatorname A,

is the smallest

convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...

containing

A.

The convex balanced hull of a set

A

is the smallest

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

balanced set containing

A.

The polar of a subset

A \subseteq X

is defined to be:

A^\circ := \left\.

while the prepolar of a subset

B \subseteq X^

is:

^ B := \left\.

The bipolar of a subset

A \subseteq X,

often denoted by

A^

is the set

A^ := ^\left(A^\right) 
= \left\.

Statement in functional analysis

Let

\sigma\left(X, X^\right)

denote the weak topology on

X

(that is, the weakest TVS topology on

X

making all linear functionals in

X^

continuous). :The bipolar theorem: The bipolar of a subset

A \subseteq X

is equal to the

\sigma\left(X, X^\right)

-closure of the convex balanced hull of

A.

Statement in convex analysis

:The bipolar theorem: For any nonempty cone

A

in some linear space

X,

the bipolar set

A^

is given by:

A^ = \operatorname (\operatorname \).

Special case

A subset

C \subseteq X

is a nonempty closed convex cone if and only if

C^ = C^ = C

when

C^ = \left(C^\right)^,

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

C^ = \operatorname C.

Relation to the Fenchel–Moreau theorem

Let

f(x) := \delta(x, C) = \begin0 & x \in C\\ \infty & \text\end

be the

indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...

for a cone

C.

Then the

convex conjugate In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformati ...

f^*(x^*) = \delta\left(x^*, C^\circ\right) = \delta^*\left(x^*, C\right) = \sup_ \langle x^*,x \rangle

is the support function for

C,

and

f^(x) = \delta(x, C^).

Therefore,

C = C^

if and only if

f = f^.

References

Bibliography

* * * {{Topological vector spaces Convex analysis Functional analysis Theorems in mathematical analysis Linear functionals