Distinguished space

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

Definition

Suppose that is a locally convex space and let and denote the strong dual of .
Let denote the continuous dual space of and let denote the strong dual of
Let denote endowed with the weak-* topology induced by where this topology is denoted by .
We say that a subset of is -bounded if it is a bounded subset of and we call the closure of in the TVS the -closure of.
If is a subset of then the polar of is
A Hausdorff locally convex space is called a distinguished space if it satisfies any of the following equivalent conditions:

If is a -bounded subset of then there exists a bounded subset of whose -closure contains.
If is a -bounded subset of then there exists a bounded subset of such that is contained in which is the polar of
The strong dual of is a barrelled space.

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

All normed spaces and semi-reflexive spaces are distinguished spaces.
LF spaces are distinguished spaces.
The strong dual space of a Fréchet space is distinguished if and only if is quasibarrelled.

Every locally convex distinguished space is an H-space.

There exist distinguished Banach spaces spaces that are not semi-reflexive.
The strong dual of a distinguished Banach space is not necessarily separable; [Lp space|] is such a space.
The strong dual space of a distinguished Fréchet space is not necessarily metrizable.
There exists a distinguished semi-reflexive non-reflexive -quasibarrelled Mackey space whose strong dual is a non-reflexive Banach space.
There exist H-spaces that are not distinguished spaces.
Fréchet Montel spaces are distinguished spaces.