Semi-reflexive space explained

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 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 (i.e. for any x \in X there exists some x ^{\prime} \in X ^{\prime} such that x ^{\prime} (x) ≠ 0 ).Let X ^{\prime} _{b} and X ^{\prime} _{\beta} both denote the strong dual of, which is the vector space X ^{\prime} of continuous linear functionals on endowed with the topology of uniform convergence on bounded subsets of ;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 is a normed space, then the strong dual of is the continuous dual space X ^{\prime} with its usual norm topology.The bidual of, denoted by X ^{\prime\prime} , is the strong dual of X ^{\prime} _{b} ; that is, it is the space \left(X ^{\prime} _{b} .

For any

x \in X, let J _{x} : X ^{\prime} \to F be defined by J ^{\prime} _{x\left(x} \right) = x ^{\prime} (x) , where J _{x} is called the evaluation map at ;since J _{x} : X ^{\prime} _{b} \to F is necessarily continuous, it follows that J _{x} \in \left(X ^{\prime} _{b\right)} .Since X ^{\prime} separates points on, the map J : X \to \left(X ^{\prime} _{b\right)} 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 semireflexive if

J : X \to \left(X ^{\prime} _{b\right)} is bijective (or equivalently, surjective) and we call reflexive if in addition J : X \to X ^{\prime\prime} = \left(X ^{\prime} _{b} is an isomorphism of TVSs.If is a normed space then is a TVS-embedding as well as an isometry onto its range;furthermore, by Goldstine's theorem (proved in 1938), the range of 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 be a topological vector space over a number field

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 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 .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} , which is called the strong bidual space for .It consists of allcontinuous linear functionals h : X ^{\prime} _{b} \to { F} and is equipped with the strong topology b\left(\left(X ^{\prime} _{b\right)} , X ^{\prime} _{b} \right) .Each vector x\in X generates a map J(x) : X ^{\prime} _{b} \to F by the following formula:

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

This is a continuous linear functional on

X ^{\prime} _{b} , that is, J(x) \in \left(X ^{\prime} _{b} .One obtains a map called the evaluation map or the canonical injection:

$$J\; :\; X\; \backslash to\; \backslash left(X^\_b\backslash right)^\_.$$

which is a linear map.If is locally convex, from the Hahn–Banach theorem it follows that is injective and open (that is, for each neighbourhood of zero

U in there is a neighbourhood of zero in \left(X ^{\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} is surjective (hence bijective); it is called reflexive if the evaluation map J : X \to \left(X ^{\prime} _{b} is surjective and continuous, in which case will be an isomorphism of TVSs).

Characterizations of semi-reflexive spaces

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

1. is semireflexive;
2. the weak topology on 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).

1. 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;X ^{\prime} _{\tau} is barrelled, where the \tau indicates the Mackey topology on X ^{\prime} ;

1. 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

See main article: Reflexive space.

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

1. is reflexive;
2. is semireflexive and barrelled;
3. is barrelled and the weak topology on 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).

1. is semireflexive and quasibarrelled.

If is a normed space then the following are equivalent:

1. is reflexive;
2. the closed unit ball is compact when has the weak topology

\sigma\left(X, X ^{\prime} \right) .

1. 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.

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

• • Book: Edwards , R. E. . 1965. Functional analysis. Theory and applications. Holt, Rinehart and Winston. New York. 0030505356.
• John B. Conway, A Course in Functional Analysis, Springer, 1985.
• .
  • Book: Kolmogorov. A. N.. Fomin. S. V.. 1957. Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces. Graylock Press. Rochester.
• .