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 ...

, distinguished spaces are

topological vector spaces 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 ...

(TVSs) having the property that weak-* bounded subsets of their biduals (that is, the

strong dual space In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...

of their strong dual space) are contained in the weak-* closure of some bounded subset of the bidual.

Definition

Suppose that

X

is 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 ...

and let

X^

and

X^_b

denote the

strong dual In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...

X

(that is, the

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

endowed with the

strong dual topology In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...

). Let

X^

denote the continuous dual space of

X^_b

and let

X^_b

denote the strong dual of

X^_b.

Let

X^_

denote

X^

endowed with 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 ...

induced by

X^,

where this topology is denoted by

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

(that is, the topology of pointwise convergence on

X^

). We say that a subset

W

X^

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

-bounded if it is a bounded subset of

X^_

and we call the closure of

W

in the TVS

X^_

the

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

-closure of

W

. If

B

is a subset of

X

then 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 ...

B

B^ := \left\.

A Hausdorff

X

is called a distinguished space if it satisfies any of the following equivalent conditions:

If $W \subseteq X^$ is a $\sigma\left(X^, X^\right)$ -bounded subset of $X^$ then there exists a bounded subset $B$ of $X^_b$ whose $\sigma\left(X^, X^\right)$ -closure contains $W$ .
If $W \subseteq X^$ is a $\sigma\left(X^, X^\right)$ -bounded subset of $X^$ then there exists a bounded subset $B$ of $X$ such that $W$ is contained in $B^ := \left\,$ which 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 ...
(relative to the duality $\left\langle X^, X^ \right\rangle$ ) of $B^.$
The
strong dual In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
of $X$ is a
barrelled space In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a ...
.

If in addition

X

is a

metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...

locally convex topological vector 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 ...

then this list may be extended to include:

( Grothendieck) The strong dual of $X$ is a
bornological space In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a ...
.

Sufficient conditions

All

normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...

s and

semi-reflexive space In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) ''X'' such that the canonical evaluation map from ''X'' into its bidual (which is the strong dual of the strong dual ...

s are distinguished spaces.

LF space In mathematics, an ''LF''-space, also written (''LF'')-space, is a topological vector space (TVS) ''X'' that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Fréchet spaces. This means that ''X'' is a direct limi ...

s are distinguished spaces. The

X_b^

of a

Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the ...

X

is distinguished if and only if

X

is quasibarrelled.Gabriyelyan, S.S
"On topological spaces and topological groups with certain local countable networks
(2014)

Properties

Every locally convex distinguished space is an

H-space In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed. Definition An H-space consists of a topological space , together wi ...

Examples

There exist distinguished

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

s spaces that are not

semi-reflexive In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) ''X'' such that the canonical evaluation map from ''X'' into its bidual (which is the strong dual of the strong dual ...

. The

of a distinguished Banach space is not necessarily separable;

l^

is such a space. The

of a distinguished

is not necessarily

. There exists a distinguished

non- reflexive - quasibarrelled

Mackey space In mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space ''X'' such that the topology of ''X'' coincides with the Mackey topology τ(''X'',''X′''), the finest topology which still pres ...

X

whose strong dual is a non-reflexive Banach space. There exist

s that are not distinguished spaces. Fréchet

Montel space In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a Barrelled space, barrelled topo ...

s are distinguished spaces.

References

Bibliography

* * * * * * * * {{Topological vector spaces Topological vector spaces