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 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.

Brief definition
Suppose that $X$ is a topological vector space (TVS) over the field $$\mathbb{F}$$ (which is either the real or complex numbers) whose continuous dual space, $$X^{\prime}$$, separates points on $X$ (i.e. for any $$x \in X$$ there exists some $$x^{\prime} \in X^{\prime}$$ such that $$x^{\prime}(x) \neq 0$$). Let $$X^{\prime}_b$$ and $$X^{\prime}_{\beta}$$ both denote the strong dual of $X$, which is the vector space $$X^{\prime}$$ 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 $$X^{\prime}$$ with its usual norm topology. The bidual of $X$, denoted by $$X^{\prime\prime}$$, is the strong dual of $$X^{\prime}_b$$; that is, it is the space $$\left(X^{\prime}_b\right)^{\prime}_{b}$$.

For any $$x \in X,$$ let $$J_x : X^{\prime} \to \mathbb{F}$$ be defined by $$J_x\left(x^{\prime}\right) = x^{\prime}(x)$$, where $$J_x$$ is called the evaluation map at $x$; since $$J_x : X^{\prime}_b \to \mathbb{F}$$ is necessarily continuous, it follows that $$J_x \in \left(X^{\prime}_b\right)^{\prime}$$. Since $$X^{\prime}$$ separates points on $X$, the map $$J : X \to \left(X^{\prime}_b\right)^{\prime}$$ defined by $$J(x) := J_x$$ 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 $$J : X \to \left(X^{\prime}_b\right)^{\prime}$$ is bijective (or equivalently, surjective) and we call $X$ reflexive if in addition $$J : X \to X^{\prime\prime} = \left(X^{\prime}_b\right)^{\prime}_b$$ 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 $$\left(X^{\prime\prime}, \sigma\left(X^{\prime\prime}, X^{\prime}\right)\right)$$. 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 $$\sigma\left(X^{\prime}, X\right)$$-compact.

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

$$J(x)(f) = f(x),\qquad f \in X'.$$

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

$$J : X \to \left(X^{\prime}_b\right)^{\prime}_{b}.$$

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 $$U$$ in $X$ there is a neighbourhood of zero $V$ in $$\left(X^{\prime}_b\right)^{\prime}_{b}$$ such that $$J(U) \supseteq V \cap J(X)$$). But it can be non-surjective and/or discontinuous.

A locally convex space $$X$$ is called semi-reflexive if the evaluation map $$J : X \to \left(X^{\prime}_b\right)^{\prime}_{b}$$ is surjective (hence bijective); it is called reflexive if the evaluation map  $$J : X \to \left(X^{\prime}_b\right)^{\prime}_{b}$$ 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 $$\sigma\left(X, X^{\prime}\right)$$, every closed and bounded subset of $$X_{\sigma}$$ is weakly compact).
 * 3) If linear form on $$X^{\prime}$$ that continuous when $$X^{\prime}$$ has the strong dual topology, then it is continuous when $$X^{\prime}$$ has the weak topology;
 * 4) $$X^{\prime}_{\tau}$$ is barrelled, where the $$\tau$$ indicates the Mackey topology on $$X^{\prime}$$;
 * 5) $X$ weak the weak topology $$\sigma\left(X, X^{\prime}\right)$$ is quasi-complete.

$$

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

Properties
If $$X$$ is a Hausdorff locally convex space then the canonical injection from $$X$$ into its bidual is a topological embedding if and only if $$X$$ 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 $$\sigma\left(X, X^{\prime}\right)$$, every closed and bounded subset of $$X_{\sigma}$$ 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 $$\sigma\left(X, X^{\prime}\right)$$.
 * 3) $X$ is a Banach space and $$X^{\prime}_b$$ is reflexive.

Examples
Every non-reflexive infinite-dimensional Banach space is a distinguished space that is not semi-reflexive. If $$X$$ is a dense proper vector subspace of a reflexive Banach space then $$X$$ 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.