View source for 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 space|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 {{mvar|X}} is a [[topological vector space]] (TVS) over the field <math>\mathbb{F}</math> (which is either the real or complex numbers) whose [[continuous dual space]], <math>X^{\prime}</math>, '''separates points''' on {{mvar|X}} (i.e. for any <math>x \in X</math> there exists some <math>x^{\prime} \in X^{\prime}</math> such that <math>x^{\prime}(x) \neq 0</math>).
Let <math>X^{\prime}_b</math> and <math>X^{\prime}_{\beta}</math> both denote the [[strong dual]] of {{mvar|X}}, which is the vector space <math>X^{\prime}</math> of continuous linear functionals on {{mvar|X}} endowed with the [[topology of uniform convergence]] on [[Topology of uniform convergence#Strong dual topology b(X*, X)|bounded subsets]] of {{mvar|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 {{mvar|X}} is a normed space, then the strong dual of {{mvar|X}} is the continuous dual space <math>X^{\prime}</math> with its usual norm topology.
The '''bidual''' of {{mvar|X}}, denoted by <math>X^{\prime\prime}</math>, is the strong dual of <math>X^{\prime}_b</math>; that is, it is the space <math>\left(X^{\prime}_b\right)^{\prime}_{b}</math>.{{sfn|Trèves|2006|pp=372–374}}

For any <math>x \in X,</math> let <math>J_x : X^{\prime} \to \mathbb{F}</math> be defined by <math>J_x\left(x^{\prime}\right) = x^{\prime}(x)</math>, where <math>J_x</math> is called the '''evaluation map at {{mvar|x}}''';
since <math>J_x : X^{\prime}_b \to \mathbb{F}</math> is necessarily continuous, it follows that <math>J_x \in \left(X^{\prime}_b\right)^{\prime}</math>.
Since <math>X^{\prime}</math> separates points on {{mvar|X}}, the map <math>J : X \to \left(X^{\prime}_b\right)^{\prime}</math> defined by <math>J(x) := J_x</math> is injective where this map is called the '''evaluation map''' or the '''canonical map'''.
This map was introduced by [[Hans Hahn (mathematician)|Hans Hahn]] in 1927.{{sfn|Narici|Beckenstein|2011|pp=225–273}}

We call {{mvar|X}} '''semireflexive''' if <math>J : X \to \left(X^{\prime}_b\right)^{\prime}</math> is bijective (or equivalently, [[surjective]]) and we call {{mvar|X}} '''[[Reflexive space|reflexive]]''' if in addition <math>J : X \to X^{\prime\prime} = \left(X^{\prime}_b\right)^{\prime}_b</math> is an isomorphism of TVSs.{{sfn|Trèves|2006|pp=372–374}}
If {{mvar|X}} is a normed space then {{mvar|J}} is a TVS-embedding as well as an [[isometry]] onto its range;
furthermore, by [[Goldstine's theorem]] (proved in 1938), the range of {{mvar|J}} is a dense subset of the bidual <math>\left(X^{\prime\prime}, \sigma\left(X^{\prime\prime}, X^{\prime}\right)\right)</math>.{{sfn|Narici|Beckenstein|2011|pp=225–273}}
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 <math>\sigma\left(X^{\prime}, X\right)</math>-compact.{{sfn|Narici|Beckenstein|2011|pp=225–273}} 

=== Detailed definition ===

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

<math display=block>J(x)(f) = f(x),\qquad f \in X'.</math>

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

<math display=block>J : X \to \left(X^{\prime}_b\right)^{\prime}_{b}.</math>

which is a linear map.
If {{mvar|X}} is locally convex, from the [[Hahn–Banach theorem]] it follows that {{mvar|J}} is injective and open (that is, for each neighbourhood of zero <math>U</math> in {{mvar|X}} there is a neighbourhood of zero {{mvar|V}} in <math>\left(X^{\prime}_b\right)^{\prime}_{b}</math> such that <math>J(U) \supseteq V \cap J(X)</math>).
But it can be non-surjective and/or discontinuous.

A locally convex space <math>X</math> is called '''semi-reflexive''' if the evaluation map  <math>J : X \to \left(X^{\prime}_b\right)^{\prime}_{b}</math> is surjective (hence bijective); it is called '''reflexive''' if the evaluation map  <math>J : X \to \left(X^{\prime}_b\right)^{\prime}_{b}</math> is surjective and continuous, in which case  {{mvar|J}} will be an [[Topological vector space|isomorphism of TVSs]]).

== Characterizations of semi-reflexive spaces ==

If {{mvar|X}} is a Hausdorff locally convex space then the following are equivalent:
# {{mvar|X}} is semireflexive;
# the weak topology on {{mvar|X}} had the Heine-Borel property (that is, for the weak topology <math>\sigma\left(X, X^{\prime}\right)</math>, every closed and bounded subset of <math>X_{\sigma}</math> is weakly compact).{{sfn|Trèves|2006|pp=372–374}}
# If linear form on <math>X^{\prime}</math> that continuous when <math>X^{\prime}</math> has the strong dual topology, then it is continuous when <math>X^{\prime}</math> has the weak topology;{{sfn|Schaefer|Wolff|1999|p=144}}
# <math>X^{\prime}_{\tau}</math> is [[Barrelled space|barrelled]], where the <math>\tau</math> indicates the [[Mackey topology]] on <math>X^{\prime}</math>;{{sfn|Schaefer|Wolff|1999|p=144}}
# {{mvar|X}} weak the weak topology <math>\sigma\left(X, X^{\prime}\right)</math> is [[quasi-complete]].{{sfn|Schaefer|Wolff|1999|p=144}}

{{Math theorem|name=Theorem{{sfn|Edwards|1965|loc=8.4.2}}|math_statement=
A locally convex Hausdorff space <math>X</math> is semi-reflexive if and only if <math>X</math> with the <math>\sigma\left(X, X^{\prime}\right)</math>-topology has the Heine–Borel property (i.e. weakly closed and bounded subsets of <math>X</math> are weakly compact).
}}

== Sufficient conditions ==

Every [[Montel space|semi-Montel space]] is semi-reflexive and every [[Montel space]] is reflexive.

== Properties ==

If <math>X</math> is a Hausdorff locally convex space then the canonical injection from <math>X</math> into its bidual is a topological embedding if and only if <math>X</math> is infrabarrelled.{{sfn|Narici|Beckenstein|2011|pp=488–491}}

The strong dual of a semireflexive space is [[Barrelled space|barrelled]]. 
Every semi-reflexive space is [[Quasi-complete space|quasi-complete]].{{sfn|Schaefer|Wolff|1999|p=144}} 
Every semi-reflexive normed space is a reflexive Banach space.{{sfn|Schaefer|Wolff|1999|p=145}} 
The strong dual of a semireflexive space is barrelled.{{sfn|Edwards|1965|loc=8.4.3}}

== Reflexive spaces ==

{{Main|Reflexive space}}

If {{mvar|X}} is a Hausdorff locally convex space then the following are equivalent:
# {{mvar|X}} is [[Reflexive space|reflexive]];
# {{mvar|X}} is semireflexive and [[barrelled space|barrelled]];
# {{mvar|X}} is barrelled and the weak topology on {{mvar|X}} had the Heine-Borel property (which means that for the weak topology <math>\sigma\left(X, X^{\prime}\right)</math>, every closed and bounded subset of <math>X_{\sigma}</math> is weakly compact).{{sfn|Trèves|2006|pp=372-374}}
# {{mvar|X}} is semireflexive and [[Quasibarrelled space|quasibarrelled]].{{sfn|Khaleelulla|1982|pp=32–63}}

If {{mvar|X}} is a [[normed space]] then the following are equivalent:
# {{mvar|X}} is reflexive;
# the closed unit ball is compact when {{mvar|X}} has the weak topology <math>\sigma\left(X, X^{\prime}\right)</math>.{{sfn|Trèves|2006|p=376}}
# {{mvar|X}} is a Banach space and <math>X^{\prime}_b</math> is reflexive.{{sfn|Trèves|2006|p=377}}

== Examples ==

Every non-[[Reflexive space|reflexive]] infinite-dimensional [[Banach space]] is a [[distinguished space]] that is not semi-reflexive.{{sfn|Khaleelulla|1982|pp=28-63}}
If <math>X</math> is a dense proper vector subspace of a reflexive Banach space then <math>X</math> is a normed space that not semi-reflexive but its strong dual space is a reflexive Banach space.{{sfn|Khaleelulla|1982|pp=28-63}}
There exists a semi-reflexive [[countably barrelled space]] that is not [[Barrelled space|barrelled]].{{sfn|Khaleelulla|1982|pp=28-63}}

== 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]]

== Citations ==
{{reflist|group=note}}
{{reflist}}

== Bibliography ==
* {{Edwards Functional Analysis Theory and Applications}} <!-- {{sfn|Edwards|1995|p=}} -->
* {{cite book
|last = Edwards
|first = R. E.
|year = 1965
|title = Functional analysis. Theory and applications
|publisher = Holt, Rinehart and Winston
|location = New York
|isbn = 0030505356
}}
* [[John B. Conway]], ''A Course in Functional Analysis'', Springer, 1985.
* {{citation
|last = James
|first = Robert C. |author-link=Robert C. James
|title = Some self-dual properties of normed linear spaces. Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967)
|pages = 159–175
|series = Ann. of Math. Studies
|volume = 69
|publisher = Princeton Univ. Press
|location = Princeton, NJ
|year = 1972
}}.
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn|Khaleelulla|1982|p=}} -->
* {{cite book
|last1 = Kolmogorov
|first1 = A. N.
|last2 = Fomin
|first2 = S. V.
|year = 1957
|title = Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces
|publisher = Graylock Press
|location = Rochester
|isbn =
}}
* {{citation
|last = Megginson|first = Robert E.|authorlink = Robert Megginson
|title = An introduction to Banach space theory
|series = Graduate Texts in Mathematics
|volume = 183
|publisher = Springer-Verlag
|location = New York
|year = 1998
|pages = xx+596
|isbn = 0-387-98431-3
}}.
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Schechter Handbook of Analysis and Its Foundations}} <!-- {{sfn|Schechter|1996|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!-- {{sfn|Wilansky|2013|p=}} -->

{{Functional analysis}}
{{TopologicalVectorSpaces}}

[[Category:Banach spaces]]
[[Category:Duality theories]]