Semi-reflexive space

In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) X such that the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is bijective. If this map is also an isomorphism of TVSs then it is called reflexive.

Semi-reflexive spaces play an important role in the general theory of locally convex TVSs. Since a normable TVS is semi-reflexive if and only if it is reflexive, the concept of semi-reflexivity is primarily used with TVSs that are not normable.

Definition and notation

Brief definition

Suppose that X is a topological vector space (TVS) over the field [math]\displaystyle{ \mathbb{F} }[/math] (which is either the real or complex numbers) whose continuous dual space, [math]\displaystyle{ X^{\prime} }[/math], separates points on X (i.e. for any [math]\displaystyle{ x \in X }[/math] there exists some [math]\displaystyle{ x^{\prime} \in X^{\prime} }[/math] such that [math]\displaystyle{ x^{\prime}(x) \neq 0 }[/math]). Let [math]\displaystyle{ X^{\prime}_b }[/math] and [math]\displaystyle{ X^{\prime}_{\beta} }[/math] both denote the strong dual of X, which is the vector space [math]\displaystyle{ X^{\prime} }[/math] of continuous linear functionals on X endowed with the topology of uniform convergence on bounded subsets of X; this topology is also called the strong dual topology and it is the "default" topology placed on a continuous dual space (unless another topology is specified). If X is a normed space, then the strong dual of X is the continuous dual space [math]\displaystyle{ X^{\prime} }[/math] with its usual norm topology. The bidual of X, denoted by [math]\displaystyle{ X^{\prime\prime} }[/math], is the strong dual of [math]\displaystyle{ X^{\prime}_b }[/math]; that is, it is the space [math]\displaystyle{ \left(X^{\prime}_b\right)^{\prime}_{b} }[/math].

For any [math]\displaystyle{ x \in X, }[/math] let [math]\displaystyle{ J_x : X^{\prime} \to \mathbb{F} }[/math] be defined by [math]\displaystyle{ J_x\left(x^{\prime}\right) = x^{\prime}(x) }[/math], where [math]\displaystyle{ J_x }[/math] is called the evaluation map at x; since [math]\displaystyle{ J_x : X^{\prime}_b \to \mathbb{F} }[/math] is necessarily continuous, it follows that [math]\displaystyle{ J_x \in \left(X^{\prime}_b\right)^{\prime} }[/math]. Since [math]\displaystyle{ X^{\prime} }[/math] separates points on X, the map [math]\displaystyle{ J : X \to \left(X^{\prime}_b\right)^{\prime} }[/math] defined by [math]\displaystyle{ J(x) := J_x }[/math] is injective where this map is called the evaluation map or the canonical map. This map was introduced by Hans Hahn in 1927.

We call X semireflexive if [math]\displaystyle{ J : X \to \left(X^{\prime}_b\right)^{\prime} }[/math] is bijective (or equivalently, surjective) and we call X reflexive if in addition [math]\displaystyle{ J : X \to X^{\prime\prime} = \left(X^{\prime}_b\right)^{\prime}_b }[/math] is an isomorphism of TVSs. If X is a normed space then J is a TVS-embedding as well as an isometry onto its range; furthermore, by Goldstine's theorem (proved in 1938), the range of J is a dense subset of the bidual [math]\displaystyle{ \left(X^{\prime\prime}, \sigma\left(X^{\prime\prime}, X^{\prime}\right)\right) }[/math]. A normable space is reflexive if and only if it is semi-reflexive. A Banach space is reflexive if and only if its closed unit ball is [math]\displaystyle{ \sigma\left(X^{\prime}, X\right) }[/math]-compact.

Detailed definition

Let X be a topological vector space over a number field [math]\displaystyle{ \mathbb{F} }[/math] (of real numbers [math]\displaystyle{ \R }[/math] or complex numbers [math]\displaystyle{ \C }[/math]). Consider its strong dual space [math]\displaystyle{ X^{\prime}_b }[/math], which consists of all continuous linear functionals [math]\displaystyle{ f : X \to \mathbb{F} }[/math] and is equipped with the strong topology [math]\displaystyle{ b\left(X^{\prime}, X\right) }[/math], that is, the topology of uniform convergence on bounded subsets in X. The space [math]\displaystyle{ X^{\prime}_b }[/math] is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space [math]\displaystyle{ \left(X^{\prime}_b\right)^{\prime}_{b} }[/math], which is called the strong bidual space for X. It consists of all continuous linear functionals [math]\displaystyle{ h : X^{\prime}_b \to {\mathbb F} }[/math] and is equipped with the strong topology [math]\displaystyle{ b\left(\left(X^{\prime}_b\right)^{\prime}, X^{\prime}_b \right) }[/math]. Each vector [math]\displaystyle{ x\in X }[/math] generates a map [math]\displaystyle{ J(x) : X^{\prime}_b \to \mathbb{F} }[/math] by the following formula:

[math]\displaystyle{ J(x)(f) = f(x),\qquad f \in X'. }[/math]

This is a continuous linear functional on [math]\displaystyle{ X^{\prime}_b }[/math], that is, [math]\displaystyle{ J(x) \in \left(X^{\prime}_b\right)^{\prime}_{b} }[/math]. One obtains a map called the evaluation map or the canonical injection:

[math]\displaystyle{ J : X \to \left(X^{\prime}_b\right)^{\prime}_{b}. }[/math]

which is a linear map. If X is locally convex, from the Hahn–Banach theorem it follows that J is injective and open (that is, for each neighbourhood of zero [math]\displaystyle{ U }[/math] in X there is a neighbourhood of zero V in [math]\displaystyle{ \left(X^{\prime}_b\right)^{\prime}_{b} }[/math] such that [math]\displaystyle{ J(U) \supseteq V \cap J(X) }[/math]). But it can be non-surjective and/or discontinuous.

A locally convex space [math]\displaystyle{ X }[/math] is called semi-reflexive if the evaluation map [math]\displaystyle{ J : X \to \left(X^{\prime}_b\right)^{\prime}_{b} }[/math] is surjective (hence bijective); it is called reflexive if the evaluation map [math]\displaystyle{ J : X \to \left(X^{\prime}_b\right)^{\prime}_{b} }[/math] is surjective and continuous, in which case J will be an isomorphism of TVSs).

Characterizations of semi-reflexive spaces

If X is a Hausdorff locally convex space then the following are equivalent:

1. X is semireflexive;
2. the weak topology on X had the Heine-Borel property (that is, for the weak topology [math]\displaystyle{ \sigma\left(X, X^{\prime}\right) }[/math], every closed and bounded subset of [math]\displaystyle{ X_{\sigma} }[/math] is weakly compact).
3. If linear form on [math]\displaystyle{ X^{\prime} }[/math] that continuous when [math]\displaystyle{ X^{\prime} }[/math] has the strong dual topology, then it is continuous when [math]\displaystyle{ X^{\prime} }[/math] has the weak topology;
4. [math]\displaystyle{ X^{\prime}_{\tau} }[/math] is barrelled, where the [math]\displaystyle{ \tau }[/math] indicates the Mackey topology on [math]\displaystyle{ X^{\prime} }[/math];
5. X weak the weak topology [math]\displaystyle{ \sigma\left(X, X^{\prime}\right) }[/math] is quasi-complete.

Theorem — A locally convex Hausdorff space [math]\displaystyle{ X }[/math] is semi-reflexive if and only if [math]\displaystyle{ X }[/math] with the [math]\displaystyle{ \sigma\left(X, X^{\prime}\right) }[/math]-topology has the Heine–Borel property (i.e. weakly closed and bounded subsets of [math]\displaystyle{ X }[/math] are weakly compact).

Sufficient conditions

Every semi-Montel space is semi-reflexive and every Montel space is reflexive.

Properties

If [math]\displaystyle{ X }[/math] is a Hausdorff locally convex space then the canonical injection from [math]\displaystyle{ X }[/math] into its bidual is a topological embedding if and only if [math]\displaystyle{ X }[/math] is infrabarrelled.

The strong dual of a semireflexive space is barrelled. Every semi-reflexive space is quasi-complete. Every semi-reflexive normed space is a reflexive Banach space. The strong dual of a semireflexive space is barrelled.

Reflexive spaces

If X is a Hausdorff locally convex space then the following are equivalent:

1. X is reflexive;
2. X is semireflexive and barrelled;
3. X is barrelled and the weak topology on X had the Heine-Borel property (which means that for the weak topology [math]\displaystyle{ \sigma\left(X, X^{\prime}\right) }[/math], every closed and bounded subset of [math]\displaystyle{ X_{\sigma} }[/math] is weakly compact).
4. X is semireflexive and quasibarrelled.

If X is a normed space then the following are equivalent:

1. X is reflexive;
2. the closed unit ball is compact when X has the weak topology [math]\displaystyle{ \sigma\left(X, X^{\prime}\right) }[/math].
3. X is a Banach space and [math]\displaystyle{ X^{\prime}_b }[/math] is reflexive.

Examples

Every non-reflexive infinite-dimensional Banach space is a distinguished space that is not semi-reflexive. If [math]\displaystyle{ X }[/math] is a dense proper vector subspace of a reflexive Banach space then [math]\displaystyle{ X }[/math] is a normed space that not semi-reflexive but its strong dual space is a reflexive Banach space. There exists a semi-reflexive countably barrelled space that is not barrelled.

See also

• Grothendieck space - A generalization which has some of the properties of reflexive spaces and includes many spaces of practical importance.
• Reflexive operator algebra
• Reflexive space

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