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.

Definition and notation

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 $X_{b}^{\prime }$ and $X_{\beta }^{\prime }$ 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 $X_{b}^{\prime }$; that is, it is the space $\left(X_{b}^{\prime }\right)_{b}^{\prime }$.

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 $J_{x}:X_{b}^{\prime }\to \mathbb {F}$ is necessarily continuous, it follows that $J_{x}\in \left(X_{b}^{\prime }\right)^{\prime }$. Since $X^{\prime }$ separates points on X, the map $J:X\to \left(X_{b}^{\prime }\right)^{\prime }$ 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 $J:X\to \left(X_{b}^{\prime }\right)^{\prime }$ is bijective (or equivalently, surjective) and we call X reflexive if in addition $J:X\to X^{\prime \prime }=\left(X_{b}^{\prime }\right)_{b}^{\prime }$ 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 $\mathbb {R}$ or complex numbers $\mathbb {C}$). Consider its strong dual space $X_{b}^{\prime }$, 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 $X_{b}^{\prime }$ is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space $\left(X_{b}^{\prime }\right)_{b}^{\prime }$, which is called the strong bidual space for X. It consists of all continuous linear functionals $h:X_{b}^{\prime }\to {\mathbb {F} }$ and is equipped with the strong topology $b\left(\left(X_{b}^{\prime }\right)^{\prime },X_{b}^{\prime }\right)$. Each vector $x\in X$ generates a map $J(x):X_{b}^{\prime }\to \mathbb {F}$ by the following formula:

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

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

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

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 $U$ in X there is a neighbourhood of zero V in $\left(X_{b}^{\prime }\right)_{b}^{\prime }$ 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 $J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }$ is surjective (hence bijective); it is called reflexive if the evaluation map $J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }$ 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. $X_{\tau }^{\prime }$ 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.

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

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

Main article: Reflexive space

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 $X_{b}^{\prime }$ 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.

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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