Distinguished space

Short description: TVS whose strong dual is barralled

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

A Hausdorff locally convex space [math]\displaystyle{ X }[/math] is called a distinguished space if it satisfies any of the following equivalent conditions:

1. If [math]\displaystyle{ W \subseteq X^{\prime \prime} }[/math] is a [math]\displaystyle{ \sigma\left(X^{\prime \prime}, X^{\prime}\right) }[/math]-bounded subset of [math]\displaystyle{ X^{\prime \prime} }[/math] then there exists a bounded subset [math]\displaystyle{ B }[/math] of [math]\displaystyle{ X^{\prime \prime}_b }[/math] whose [math]\displaystyle{ \sigma\left(X^{\prime \prime}, X^{\prime}\right) }[/math]-closure contains [math]\displaystyle{ W }[/math].
2. If [math]\displaystyle{ W \subseteq X^{\prime \prime} }[/math] is a [math]\displaystyle{ \sigma\left(X^{\prime \prime}, X^{\prime}\right) }[/math]-bounded subset of [math]\displaystyle{ X^{\prime \prime} }[/math] then there exists a bounded subset [math]\displaystyle{ B }[/math] of [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ W }[/math] is contained in [math]\displaystyle{ 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\}, }[/math] which is the polar (relative to the duality [math]\displaystyle{ \left\langle X^{\prime}, X^{\prime\prime} \right\rangle }[/math]) of [math]\displaystyle{ B^{\circ}. }[/math]
3. The strong dual of [math]\displaystyle{ X }[/math] is a barrelled space.

If in addition [math]\displaystyle{ X }[/math] is a metrizable locally convex topological vector space then this list may be extended to include:

4. (Grothendieck) The strong dual of [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X_b^{\prime} }[/math] of a Fréchet space [math]\displaystyle{ X }[/math] is distinguished if and only if [math]\displaystyle{ X }[/math] 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; [math]\displaystyle{ l^{1} }[/math] 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 [math]\displaystyle{ X }[/math] 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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