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 is contained in the weak_* closure of some bounded subset of the bidual.

Definition

Suppose that X is a locally convex space and let and denote the strong dual of X.
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 W of is -bounded if it is a bounded subset of and we call the closure of W in the TVS the -closure of W.
If B is a subset of X then the polar of B is.
A Hausdorff locally convex TVS X is called a distinguished space if it satisfies any of the following equivalent conditions:

  1. If W ⊆ is a -bounded subset of then there exists a bounded subset B of whose -closure contains W.
  2. If W ⊆ is a -bounded subset of then there exists a bounded subset B of X such that W is contained in, which is the polar of.
  3. The strong dual of X is a barrelled space.
If in addition X is a metrizable locally convex space then we may add to this list:

  1. The strong dual of X is a bornological space.

Sufficient conditions

Every normed space and semi-reflexive space is a distinguished space.

Properties

Every locally convex distinguished space is an H-space.

Examples

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 of a distinguished Fréchet space is not necessarily metrizable.
There exists a distinguished semi-reflexive non-reflexive non-quasibarrelled Mackey space X whose strong dual is a non-reflexive Banach space.
There exist H-spaces that are not distinguished spaces.