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 ${X_b}^{\prime }$ 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 ${X_b}^{\prime }$ and let ${X_b}^{″}$ denote the strong dual of ${X_b}^{\prime }.$ Let $X_{{\sigma }^{″}}$ denote $X^{″}$ endowed with the weak-* topology induced by $X^{\prime },$ where this topology is denoted by $\sigma (X^{″},X^{\prime })$ (that is, the topology of pointwise convergence on $X^{\prime }$). We say that a subset $W$ of $X^{″}$ is $\sigma (X^{″},X^{\prime })$-bounded if it is a bounded subset of $X_{{\sigma }^{″}}$ and we call the closure of $W$ in the TVS $X_{{\sigma }^{″}}$ the $\sigma (X^{″},X^{\prime })$-closure of $W$. If $B$ is a subset of $X$ then the polar of $B$ is $B^{\circ }:=\{x^{\prime }\in X^{\prime }:\underset{b\in B}{sup}⟨b,x^{\prime }⟩\le 1\}.$

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 $\sigma (X^{″},X^{\prime })$-bounded subset of $X^{″}$ then there exists a bounded subset $B$ of ${X_b}^{″}$ whose $\sigma (X^{″},X^{\prime })$-closure contains $W$.Template:Sfn
2. If $W\subseteq X^{″}$ is a $\sigma (X^{″},X^{\prime })$-bounded subset of $X^{″}$ then there exists a bounded subset $B$ of $X$ such that $W$ is contained in $B^{\circ \circ }:=\{x^{″}\in X^{″}:\underset{x^{\prime }\in B^{\circ }}{sup}⟨x^{\prime },x^{″}⟩\le 1\},$ which is the polar (relative to the duality $⟨X^{\prime },X^{″}⟩$) of $B^{\circ }.$Template:Sfn
3. The strong dual of $X$ is a barrelled space.Template:Sfn

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.Template:Sfn

Sufficient conditions

All normed spaces and semi-reflexive spaces are distinguished spaces.Template:Sfn 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.Template:Sfn

Examples

There exist distinguished Banach spaces spaces that are not semi-reflexive.Template:Sfn The strong dual of a distinguished Banach space is not necessarily separable; $l^1$ is such a space.Template:Sfn The strong dual space of a distinguished Fréchet space is not necessarily metrizable.Template:Sfn There exists a distinguished semi-reflexive non-reflexive Template:Em-quasibarrelled Mackey space $X$ whose strong dual is a non-reflexive Banach space.Template:Sfn There exist H-spaces that are not distinguished spaces.Template:Sfn

Fréchet Montel spaces are distinguished spaces.

See also

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Bibliography

• Template:Cite journal
• Template:Robertson Topological Vector Spaces
• Template:Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces
• Template:Jarchow Locally Convex Spaces
• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels