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 ...

and related areas of

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 ...

a dual topology is a

locally convex topology In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...

on a

vector 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 ...

that is induced by the

continuous dual 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 const ...

of the vector space, by means of the

bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...

(also called ''pairing'') associated with the

dual pair In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non-degenerate bilinear map b : X \times Y \to \mathbb. Duality theory, the study of dual ...

. The different dual topologies for a given dual pair are characterized by the Mackey–Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology. Several topological properties depend only on the

and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one.

Definition

Given a

(X, Y, \langle , \rangle)

, a dual topology on

X

is a

\tau

so that :

(X, \tau)' \simeq Y.

Here

(X, \tau)'

denotes the

(X,\tau)

and

(X, \tau)' \simeq Y

means that there is a

linear isomorphism In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

\Psi : Y \to (X, \tau)',\quad y \mapsto (x \mapsto \langle x, y\rangle).

(If a locally convex topology

\tau

X

is not a dual topology, then either

\Psi

is not surjective or it is ill-defined since the linear functional

x \mapsto \langle x, y\rangle

is not continuous on

X

for some

y

Properties

* Theorem (by Mackey): Given a dual pair, the

bounded set :''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. In mathematical analysis and related areas of mat ...

s under any dual topology are identical. * Under any dual topology the same sets are barrelled.

Characterization of dual topologies

The Mackey–Arens theorem, named after

George Mackey George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry. Career Mackey earned his bachelor of arts at Rice Univer ...

and Richard Arens, characterizes all possible dual topologies on a

locally convex space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...

. The theorem shows that the coarsest dual topology is 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 ...

, the topology of uniform convergence on all finite subsets of

X'

, and the

finest topology In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as th ...

is the

Mackey topology In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey topology does not mak ...

, the topology of uniform convergence on all

absolutely convex In mathematics, a subset ''C'' of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull ...

weakly compact subsets of

X'

Mackey–Arens theorem

Given a

(X, X')

with

X

a locally convex space and

X'

its

, then

\tau

is a dual topology on

X

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

it is a

topology of uniform convergence In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves. The arti ...

on a family of

and weakly compact subsets of

X'

References

* {{DEFAULTSORT:Dual Topology Topology of function spaces

Definition

Properties

Characterization of dual topologies

Mackey–Arens theorem

See also

References