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.

Suppose that X is a topological vector space (TVS) over the field ${\displaystyle \mathbb {F} }$ (which is either the real or complex numbers) whose continuous dual space, ${\displaystyle X^{\prime }}$, separates points on X (i.e. for any ${\displaystyle x\in X}$ there exists some ${\displaystyle x^{\prime }\in X^{\prime }}$ such that ${\displaystyle x^{\prime }(x)\neq 0}$). Let ${\displaystyle X_{b}^{\prime }}$ and ${\displaystyle X_{\beta }^{\prime }}$ both denote the strong dual of X, which is the vector space ${\displaystyle X^{\prime }}$ 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 ${\displaystyle X^{\prime }}$ with its usual norm topology. The bidual of X, denoted by ${\displaystyle X^{\prime \prime }}$, is the strong dual of ${\displaystyle X_{b}^{\prime }}$; that is, it is the space ${\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }}$.¹

For any ${\displaystyle x\in X,}$ let ${\displaystyle J_{x}:X^{\prime }\to \mathbb {F} }$ be defined by ${\displaystyle J_{x}\left(x^{\prime }\right)=x^{\prime }(x)}$, where ${\displaystyle J_{x}}$ is called the evaluation map at x; since ${\displaystyle J_{x}:X_{b}^{\prime }\to \mathbb {F} }$ is necessarily continuous, it follows that ${\displaystyle J_{x}\in \left(X_{b}^{\prime }\right)^{\prime }}$. Since ${\displaystyle X^{\prime }}$ separates points on X, the map ${\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}$ defined by ${\displaystyle J(x):=J_{x}}$ is injective where this map is called the evaluation map or the canonical map. This map was introduced by Hans Hahn in 1927.²

We call X semireflexive if ${\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}$ is bijective (or equivalently, surjective) and we call X reflexive if in addition ${\displaystyle J:X\to X^{\prime \prime }=\left(X_{b}^{\prime }\right)_{b}^{\prime }}$ is an isomorphism of TVSs.¹ 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 ${\displaystyle \left(X^{\prime \prime },\sigma \left(X^{\prime \prime },X^{\prime }\right)\right)}$.² 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 ${\displaystyle \sigma \left(X^{\prime },X\right)}$-compact.²

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

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

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

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

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

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

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

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

If ${\displaystyle X}$ is a Hausdorff locally convex space then the canonical injection from ${\displaystyle X}$ into its bidual is a topological embedding if and only if ${\displaystyle X}$ is infrabarrelled.⁵

The strong dual of a semireflexive space is barrelled. Every semi-reflexive space is quasi-complete.³ Every semi-reflexive normed space is a reflexive Banach space.⁶ The strong dual of a semireflexive space is barrelled.⁷

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 ${\displaystyle \sigma \left(X,X^{\prime }\right)}$, every closed and bounded subset of ${\displaystyle X_{\sigma }}$ is weakly compact).¹
4. X is semireflexive and quasibarrelled.⁸

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 ${\displaystyle \sigma \left(X,X^{\prime }\right)}$.⁹
3. X is a Banach space and ${\displaystyle X_{b}^{\prime }}$ is reflexive.¹⁰

Every non-reflexive infinite-dimensional Banach space is a distinguished space that is not semi-reflexive.¹¹ If ${\displaystyle X}$ is a dense proper vector subspace of a reflexive Banach space then ${\displaystyle X}$ is a normed space that not semi-reflexive but its strong dual space is a reflexive Banach space.¹¹ There exists a semi-reflexive countably barrelled space that is not barrelled.¹¹