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 $X^$ and $X^_b$ denote the strong dual of $X$ (that is, the continuous dual space of $X$ endowed with the strong dual topology).
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 $W$ of $X^$ is $\sigma\left(X^, X^\right)$-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 $B$ is $B^ := \left\.$

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

1. 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 $W$.


2. 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 $\left\langle X^, X^ \right\rangle$) of $B^.$


3. 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:

4. ( 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 $X_b^$ of a Fréchet space $X$ is distinguished if and only if $X$ is quasibarrelled.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 spaces spaces that are not semi-reflexive.
The strong dual of a distinguished Banach space is not necessarily separable; $l^$ 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.

See also

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References

Bibliography

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{{Topological vector spaces

Topological vector spaces