Distinguished space

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 $$X$$ is a locally convex space and let $$X^{\prime}$$ and $$X^{\prime}_b$$ denote the strong dual of $$X$$ (that is, the continuous dual space of $$X$$ endowed with the strong dual topology). Let $$X^{\prime \prime}$$ denote the continuous dual space of $$X^{\prime}_b$$ and let $$X^{\prime \prime}_b$$ denote the strong dual of $$X^{\prime}_b.$$ Let $$X^{\prime \prime}_{\sigma}$$ denote $$X^{\prime \prime}$$ endowed with the weak-* topology induced by $$X^{\prime},$$ where this topology is denoted by $$\sigma\left(X^{\prime \prime}, X^{\prime}\right)$$ (that is, the topology of pointwise convergence on $$X^{\prime}$$). We say that a subset $$W$$ of $$X^{\prime \prime}$$ is $$\sigma\left(X^{\prime \prime}, X^{\prime}\right)$$-bounded if it is a bounded subset of $$X^{\prime \prime}_{\sigma}$$ and we call the closure of $$W$$ in the TVS $$X^{\prime \prime}_{\sigma}$$ the $$\sigma\left(X^{\prime \prime}, X^{\prime}\right)$$-closure of $$W$$. If $$B$$ is a subset of $$X$$ then the polar of $$B$$ is $$B^{\circ} := \left\{ x^{\prime} \in X^{\prime} : \sup_{b \in B} \left\langle b, x^{\prime} \right\rangle \leq 1 \right\}.$$

A Hausdorff locally convex space $$X$$ is called a distinguished space if it satisfies any of the following equivalent conditions:

 If $$W \subseteq X^{\prime \prime}$$ is a $$\sigma\left(X^{\prime \prime}, X^{\prime}\right)$$-bounded subset of $$X^{\prime \prime}$$ then there exists a bounded subset $$B$$ of $$X^{\prime \prime}_b$$ whose $$\sigma\left(X^{\prime \prime}, X^{\prime}\right)$$-closure contains $$W$$. If $$W \subseteq X^{\prime \prime}$$ is a $$\sigma\left(X^{\prime \prime}, X^{\prime}\right)$$-bounded subset of $$X^{\prime \prime}$$ then there exists a bounded subset $$B$$ of $$X$$ such that $$W$$ is contained in $$B^{\circ\circ} := \left\{ x^{\prime\prime} \in X^{\prime\prime} : \sup_{x^{\prime} \in B^{\circ}} \left\langle x^{\prime}, x^{\prime\prime} \right\rangle \leq 1 \right\},$$ which is the polar (relative to the duality $$\left\langle X^{\prime}, X^{\prime\prime} \right\rangle$$) of $$B^{\circ}.$$ The strong dual of $$X$$ is a barrelled space. 

If in addition $$X$$ 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.

The strong dual space $$X_b^{\prime}$$ of a Fréchet space $$X$$ is distinguished if and only if $$X$$ is quasibarrelled.

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; $l^{1}$ 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 $$X$$ whose strong dual is a non-reflexive Banach space. There exist H-spaces that are not distinguished spaces.

Fréchet Montel spaces are distinguished spaces.