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 $X^{\prime \prime }$ denote the continuous dual space of $X_{b}^{\prime }$ and let $X_{b}^{\prime \prime }$ denote the strong dual of $X_{b}^{\prime }.$ Let $X_{\sigma }^{\prime \prime }$ 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 $X_{\sigma }^{\prime \prime }$ and we call the closure of $W$ in the TVS $X_{\sigma }^{\prime \prime }$ 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:

1. 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 $X_{b}^{\prime \prime }$ whose $\sigma \left(X^{\prime \prime },X^{\prime }\right)$-closure contains $W$.
2. 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 $B^{\circ }.$
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 $X_{b}^{\prime }$ 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 non-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

• Montel space – Barrelled space where closed and bounded subsets are compact
• Semi-reflexive space

Bibliography

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