Distinguished space explained

In functional analysis and related areas of mathematics, distinguished spaces are topological vector spaces (TVSs) having the property that weak-* bounded subsets of their biduals (that is, the strong dual space of their strong dual space) are contained in the weak-* closure of some bounded subset of the bidual.

Definition

Suppose that

is a locally convex space and let

X^\prime

and

	\prime
X
	b

denote the strong dual of

(that is, the continuous dual space of

endowed with the strong dual topology). Let

X^\prime

denote the continuous dual space of

	\prime
X
	b

and let

	\prime\prime
X
	b

denote the strong dual of

	\prime
X
	b.

Let

	\prime\prime
X
	\sigma

denote

X^\prime

endowed with the weak-* topology induced by

X^\prime,

where this topology is denoted by

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

(that is, the topology of pointwise convergence on

X^\prime

). We say that a subset

X^\prime

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

-bounded if it is a bounded subset of

	\prime\prime
X
	\sigma

and we call the closure of

in the TVS

	\prime\prime
X
	\sigma

the

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

-closure of

. If

is a subset of

then the polar of

B^\circ:=\left\{x^\prime\inX^\prime:\sup_b\left\langleb,x^\prime\right\rangle\leq1\right\}.

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

If

W\subseteqX^\prime

is a
\sigma\left(X^\prime,X^\prime\right)

-bounded subset of
X^\prime

then there exists a bounded subset
B

of
\prime\prime
X
b

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

-closure contains
W

.
If
W\subseteqX^\prime

is a
\sigma\left(X^\prime,X^\prime\right)

-bounded subset of
X^\prime

then there exists a bounded subset
B

of
X

such that
W

is contained in
B^\circ\circ:=\left\{x^\prime\prime\inX^\prime\prime:

\sup
x^\prime\inB^\circ

\left\langlex^\prime,x^\prime\prime\right\rangle\leq1\right\},

which is the polar (relative to the duality
\left\langleX^\prime,X^\prime\prime\right\rangle

) of
B^\circ.
The strong dual of
X

is a barrelled space.

If in addition

is a metrizable locally convex topological vector space then this list may be extended to include:

(Grothendieck) The strong dual of

X

is a bornological space.

Sufficient conditions

All normed spaces and semi-reflexive spaces are distinguished spaces. LF spaces are distinguished spaces.

	\prime
X
	b

of a Fréchet space

is distinguished if and only if

is quasibarrelled.^[1]

Properties

Every locally convex distinguished space is an H-space.

Examples

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

Fréchet Montel spaces are distinguished spaces.

Bibliography

Bourbaki. Nicolas. Nicolas Bourbaki. Annales de l'Institut Fourier. French. 0042609. 5–16 (1951). Sur certains espaces vectoriels topologiques. 2. 1950. 10.5802/aif.16. free.

Notes and References

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

\sup
	x^\prime\inB^\circ