mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the bipolar theorem is a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

that characterizes the bipolar (that is, the

polar Polar may refer to: Geography Polar may refer to: * Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates * Polar climate, the c ...

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 minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of s ...

, 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 A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...

to be equal to its bipolar. The bipolar theorem can be seen as a special case of the

Fenchel–Moreau theorem In convex analysis, the Fenchel–Moreau theorem (named after Werner Fenchel and Jean Jacques Moreau) or Fenchel biconjugation theorem (or just biconjugation theorem) is a theorem which gives necessary and sufficient conditions for a function to b ...

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 In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...

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 or 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 subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...

containing

A.

The

convex balanced hull In mathematics, a subset ''C'' of a Real number, real or Complex number, complex vector space is said to be absolutely convex or disked if it is Convex set, convex and Balanced set, balanced (some people use the term "circled" instead of "balanced") ...

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 polytope ...

balanced In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...

set containing

A.

The

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 In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...

X

(that is, the weakest TVS topology on

A

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

A.

Statement in convex analysis

:The bipolar theorem: For any

nonempty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...

cone

A

in some

linear space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

X,

the bipolar set

A^

is given by:

A^ = \operatorname (\operatorname \).

Special case

A subset

C \subseteq X

is a nonempty closed

convex cone In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . ...

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 In convex analysis, the Fenchel–Moreau theorem (named after Werner Fenchel and Jean Jacques Moreau) or Fenchel biconjugation theorem (or just biconjugation theorem) is a theorem which gives necessary and sufficient conditions for a function to b ...

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 , one has \mathbf_(x)=1 if x\i ...

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 transformation ...

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

is the

support function In mathematics, the support function ''h'A'' of a non-empty closed convex set ''A'' in \mathbb^n describes the (signed) distances of supporting hyperplanes of ''A'' from the origin. The support function is a convex function on \mathbb^n. Any n ...

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 analysis