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 Template:Mvar is a topological vector space (TVS) over the field $𝔽$ (which is either the real or complex numbers) whose continuous dual space, $X^{\prime }$, separates points on Template:Mvar (i.e. for any $x\in X$ there exists some $x^{\prime }\in X^{\prime }$ such that $x^{\prime }(x)\ne 0$). Let ${X_b}^{\prime }$ and $X_{{\beta }^{\prime }}$ both denote the strong dual of Template:Mvar, which is the vector space $X^{\prime }$ of continuous linear functionals on Template:Mvar endowed with the topology of uniform convergence on bounded subsets of Template:Mvar; 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 Template:Mvar is a normed space, then the strong dual of Template:Mvar is the continuous dual space $X^{\prime }$ with its usual norm topology. The bidual of Template:Mvar, denoted by $X^{″}$, is the strong dual of ${X_b}^{\prime }$; that is, it is the space ${\left({X_b}^{\prime }\right)}_{b^{\prime }}$.Template:Sfn

For any $x\in X,$ let $J_x:X^{\prime }\to 𝔽$ be defined by $J_x\left(x^{\prime }\right)=x^{\prime }(x)$, where $J_x$ is called the evaluation map at Template:Mvar; since $J_x:{X_b}^{\prime }\to 𝔽$ is necessarily continuous, it follows that $J_x\in {\left({X_b}^{\prime }\right)}^{\prime }$. Since $X^{\prime }$ separates points on Template:Mvar, 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.Template:Sfn

We call Template:Mvar semireflexive if $J:X\to {\left({X_b}^{\prime }\right)}^{\prime }$ is bijective (or equivalently, surjective) and we call Template:Mvar reflexive if in addition $J:X\to X^{″}={{\left({X_b}^{\prime }\right)}_b}^{\prime }$ is an isomorphism of TVSs.Template:Sfn If Template:Mvar is a normed space then Template:Mvar is a TVS-embedding as well as an isometry onto its range; furthermore, by Goldstine's theorem (proved in 1938), the range of Template:Mvar is a dense subset of the bidual $(X^{″},\sigma (X^{″},X^{\prime }))$.Template:Sfn 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 $\sigma (X^{\prime },X)$-compact.Template:Sfn

Detailed definition

Let Template:Mvar be a topological vector space over a number field $𝔽$ (of real numbers $ℝ$ or complex numbers $ℂ$). Consider its strong dual space ${X_b}^{\prime }$, which consists of all continuous linear functionals $f:X\to 𝔽$ and is equipped with the strong topology $b(X^{\prime },X)$, that is, the topology of uniform convergence on bounded subsets in Template:Mvar. 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 Template:Mvar. It consists of all continuous linear functionals $h:{X_b}^{\prime }\to 𝔽$ and is equipped with the strong topology $b({\left({X_b}^{\prime }\right)}^{\prime },{X_b}^{\prime })$. Each vector $x\in X$ generates a map $J(x):{X_b}^{\prime }\to 𝔽$ by the following formula:

$$J(x)(f)=f(x),f\in X^{\prime }.$$

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 Template:Mvar is locally convex, from the Hahn–Banach theorem it follows that Template:Mvar is injective and open (that is, for each neighbourhood of zero $U$ in Template:Mvar there is a neighbourhood of zero Template:Mvar 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 Template:Mvar will be an isomorphism of TVSs).

Characterizations of semi-reflexive spaces

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

1. Template:Mvar is semireflexive;
2. the weak topology on Template:Mvar had the Heine-Borel property (that is, for the weak topology $\sigma (X,X^{\prime })$, every closed and bounded subset of $X_{\sigma }$ is weakly compact).Template:Sfn
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;Template:Sfn
4. $X_{{\tau }^{\prime }}$ is barrelled, where the $\tau$ indicates the Mackey topology on $X^{\prime }$;Template:Sfn
5. Template:Mvar weak the weak topology $\sigma (X,X^{\prime })$ is quasi-complete.Template:Sfn

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.Template:Sfn

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

Reflexive spaces

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

1. Template:Mvar is reflexive;
2. Template:Mvar is semireflexive and barrelled;
3. Template:Mvar is barrelled and the weak topology on Template:Mvar had the Heine-Borel property (which means that for the weak topology $\sigma (X,X^{\prime })$, every closed and bounded subset of $X_{\sigma }$ is weakly compact).Template:Sfn
4. Template:Mvar is semireflexive and quasibarrelled.Template:Sfn

If Template:Mvar is a normed space then the following are equivalent:

1. Template:Mvar is reflexive;
2. the closed unit ball is compact when Template:Mvar has the weak topology $\sigma (X,X^{\prime })$.Template:Sfn
3. Template:Mvar is a Banach space and ${X_b}^{\prime }$ is reflexive.Template:Sfn

Examples

Every non-reflexive infinite-dimensional Banach space is a distinguished space that is not semi-reflexive.Template:Sfn 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.Template:Sfn There exists a semi-reflexive countably barrelled space that is not barrelled.Template:Sfn

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

Bibliography

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