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, norm, topology, etc.) and the linear functions defined ...

and related areas of 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 al ...

(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 su ...

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 ...

and let

X^

and

X^_b

denote the strong dual of

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 con ...

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 s ...

). 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 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 of

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 (relative to the
duality Duality may refer to: Mathematics * Duality (mathematics), a mathematical concept ** Dual (category theory), a formalization of mathematical duality ** Duality (optimization) ** Duality (order theory), a concept regarding binary relations ** Dual ...
$\left\langle X^, X^ \right\rangle$ ) of $B^.$
The strong dual of $X$ is a barrelled space.

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 , \inf ...

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 topologica ...

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 ...
.

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 "lengt ...

s and semi-reflexive spaces are distinguished spaces. LF spaces 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 ...

X

is distinguished if and only if

X

quasibarrelled In functional analysis and related areas of mathematics, quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a neighbourhood of the origin. Quasibarrelled spaces are studied because ...

.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.

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 ve ...

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 strong dual 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 -

Mackey space

X

whose strong dual is a non-reflexive Banach space. There exist H-spaces that are not distinguished spaces. Fréchet Montel spaces are distinguished spaces.

References

Bibliography

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