Editing Reflexive space (section)

== Reflexive locally convex spaces ==

The notion of reflexive Banach space can be generalized to [[topological vector space]]s in the following way.

Let <math>X</math> be a topological vector space over a number field <math>\mathbb F</math> (of [[real number]]s <math>\mathbb R</math> or [[complex number]]s <math>\Complex</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 <math>X.</math> 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 <math>X.</math> 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^{\prime}.</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> This induces a map called the '''evaluation map''':
<math display="block">J : X \to \left(X^{\prime}_b\right)^{\prime}_b.</math>
This map is linear. If <math>X</math> is locally convex, from the [[Hahn–Banach theorem]] it follows that <math>J</math> is injective and open (that is, for each neighbourhood of zero <math>U</math> in <math>X</math> there is a neighbourhood of zero <math>V</math> 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),
* '''reflexive''' if the evaluation map  <math>J : X \to \left(X^{\prime}_b\right)^{\prime}_b</math> is surjective and continuous (in this case  <math>J</math> is an isomorphism of topological vector spaces<ref name="isomorphism">An {{em|[[TVS-isomorphism|isomorphism of topological vector spaces]]}} is a [[Linear map|linear]] and a [[Homeomorphism|homeomorphic]] map <math>\varphi : X \to Y.</math></ref>).

{{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(X, X^*)</math>-topology has the Heine–Borel property (i.e. weakly closed and bounded subsets of <math>X</math> are weakly compact).
}}

{{Math theorem|name=Theorem{{sfn|Schaefer|1966|loc=5.6, 5.5 }}{{sfn|Edwards|1965|loc=8.4.5}}|math_statement=A locally convex space <math>X</math> is reflexive if and only if it is semi-reflexive and [[Barrelled space|barreled]].
}}

{{Math theorem|name=Theorem{{sfn|Edwards|1965|loc=8.4.3}}|math_statement=
The strong dual of a semireflexive space is barrelled.
}}

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=488-491}}|math_statement=
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 [[Infrabarreled space|infrabarreled]].
}}

=== Semireflexive spaces ===
{{Main|Semi-reflexive space}}

==== Characterizations ====

If <math>X</math> is a Hausdorff locally convex space then the following are equivalent:
#<math>X</math> is semireflexive;
#The weak topology on <math>X</math> 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 barreled;{{sfn|Schaefer|Wolff|1999|p=144}} 
#<math>X</math> with the weak topology <math>\sigma\left(X, X^{\prime}\right)</math> is [[quasi-complete]].{{sfn|Schaefer|Wolff|1999|p=144}}

=== Characterizations of reflexive spaces ===

If <math>X</math> is a Hausdorff locally convex space then the following are equivalent:
#<math>X</math> is reflexive;
#<math>X</math> is [[Semireflexive space|semireflexive]] and [[Infrabarreled space|infrabarreled]];{{sfn|Narici|Beckenstein|2011|pp=488-491}}
#<math>X</math> is [[Semireflexive space|semireflexive]] and [[barreled space|barreled]];
#<math>X</math> is [[barreled space|barreled]] and the weak topology on <math>X</math> 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}}
#<math>X</math> is [[Semireflexive space|semireflexive]] and [[Quasibarrelled space|quasibarrelled]].{{sfn|Khaleelulla|1982|pp=32-63}}

If <math>X</math> is a normed space then the following are equivalent:
#<math>X</math> is reflexive;
#The closed unit ball is compact when <math>X</math> has the weak topology <math>\sigma\left(X, X^{\prime}\right).</math>{{sfn|Trèves|2006|p=376}}
#<math>X</math> is a Banach space and <math>X^{\prime}_b</math> is reflexive.{{sfn|Trèves|2006|p=377}}
#Every sequence <math>\left(C_n\right)_{n=1}^{\infty},</math> with <math>C_{n+1} \subseteq C_n</math> for all <math>n</math> of nonempty closed bounded convex subsets of <math>X</math> has nonempty intersection.{{sfn|Bernardes| 2012|p=}}

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=212}}|math_statement=
A real Banach space is reflexive if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be [[Hahn–Banach theorem|strictly separated by a hyperplane]].
}}

{{Math theorem|name=[[James's theorem|James' theorem]]|math_statement=
A [[Banach space]] <math>B</math> is reflexive if and only if every [[Continuous function|continuous]] [[linear functional]] on <math>B</math> attains its [[supremum]] on the closed [[unit ball]] in <math>B.</math>
}}

=== Sufficient conditions ===

;Normed spaces

A normed space that is semireflexive is a reflexive Banach space.{{sfn|Trèves|2006|p=375}}
A closed vector subspace of a reflexive Banach space is reflexive.{{sfn|Narici|Beckenstein|2011|pp=488-491}}

Let <math>X</math> be a Banach space and <math>M</math> a closed vector subspace of <math>X.</math> If two of <math>X, M,</math> and <math>X / M</math> are reflexive then they all are.{{sfn|Narici|Beckenstein|2011|pp=488-491}} This is why reflexivity is referred to as a {{em|three-space property}}.{{sfn|Narici|Beckenstein|2011|pp=488-491}}

;Topological vector spaces

If a [[barreled space|barreled]] locally convex Hausdorff space is semireflexive then it is reflexive.{{sfn|Trèves|2006|pp=372-374}}

The strong dual of a reflexive space is reflexive.{{sfn|Schaefer|Wolff|1999|p=145}}Every [[Montel space]] is reflexive.{{sfn|Trèves|2006|p=376}} And the strong dual of a [[Montel space]] is a Montel space (and thus is reflexive).{{sfn|Trèves|2006|p=376}}

=== Properties ===

A locally convex Hausdorff reflexive space is [[Barrelled space|barrelled]].
If <math>X</math> is a normed space then <math>I : X \to X^{\prime \prime}</math> is an isometry onto a closed subspace of <math>X^{\prime \prime}.</math>{{sfn|Trèves|2006|p=375}} This isometry can be expressed by:
<math display="block">\|x\| = \sup_{\stackrel{x^{\prime} \in X^{\prime},}{\| x^{\prime} \| \leq 1}} \left|\left\langle x^{\prime}, x \right\rangle\right|.</math>

Suppose that <math>X</math> is a normed space and <math>X^{\prime\prime}</math> is its bidual equipped with the bidual norm. Then the unit ball of <math>X,</math> 
<math>I(\{ x \in X : \|x\| \leq 1 \})</math> 
is dense in the unit ball 
<math>\left\{ x^{\prime\prime} \in X^{\prime\prime} : \left\|x^{\prime\prime}\right\| \leq 1 \right\}</math> 
of <math>X^{\prime\prime}</math> for the weak topology <math>\sigma\left(X^{\prime\prime}, X^{\prime}\right).</math>{{sfn|Trèves|2006|p=375}}

=== Examples ===

<ol>
<li> Every finite-dimensional Hausdorff [[topological vector space]] is reflexive, because <math>J</math> is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.</li>
<li> A normed space <math>X</math> is reflexive as a normed space if and only if it is reflexive as a locally convex space. This follows from the fact that for a normed space <math>X</math> its dual normed space <math>X^{\prime}</math> coincides as a topological vector space with the strong dual space <math>X^{\prime}_b.</math> As a corollary, the evaluation map <math>J : X \to X^{\prime\prime}</math> coincides with the evaluation map <math>J : X \to \left(X^{\prime}_b\right)^{\prime}_b,</math> and the following conditions become equivalent:
<ol type="i">
<li><math>X</math> is a reflexive normed space (that is, <math>J : X \to X^{\prime\prime}</math> is an isomorphism of normed spaces),</li>
<li><math>X</math> is a reflexive locally convex space (that is, <math>J : X \to \left(X^{\prime}_b\right)^{\prime}_b</math> is an isomorphism of topological vector spaces<ref name=isomorphism />),</li>
<li><math>X</math> is a semi-reflexive locally convex space (that is, <math>J : X \to \left(X^{\prime}_b\right)^{\prime}_b</math> is surjective).</li>
</ol>
</li>
<li>A (somewhat artificial) example of a semi-reflexive space that is not reflexive is obtained as follows: let <math>Y</math> be an infinite dimensional reflexive Banach space, and let <math>X</math> be the topological vector space <math>\left(Y, \sigma\left(Y, Y^{\prime}\right)\right),</math> that is, the vector space <math>Y</math> equipped with the weak topology.  Then the continuous dual of <math>X</math> and <math>Y^{\prime}</math> are the same set of functionals, and bounded subsets of <math>X</math> (that is, weakly bounded subsets of <math>Y</math>) are norm-bounded, hence the Banach space <math>Y^{\prime}</math> is the strong dual of <math>X.</math> Since <math>Y</math> is reflexive, the continuous dual of <math>X^{\prime} = Y^{\prime}</math> is equal to the image <math>J(X)</math> of <math>X</math> under the canonical embedding <math>J,</math>  but the topology on <math>X</math> (the weak topology of <math>Y</math>) is not the strong topology <math>\beta\left(X, X^{\prime}\right),</math> that is equal to the norm topology of <math>Y.</math></li>
<li>[[Montel space]]s are reflexive locally convex topological vector spaces. In particular, the following functional spaces frequently used in functional analysis are reflexive locally convex spaces:<ref>{{harvnb|Edwards|1965|loc=8.4.7}}.</ref>
* the space <math>C^\infty(M)</math> of smooth functions on arbitrary (real) smooth manifold <math>M,</math> and its strong dual space <math>\left(C^\infty\right)^{\prime}(M)</math> of distributions with compact support on <math>M,</math>
* the space <math>\mathcal{D}(M)</math> of smooth functions with compact support on arbitrary (real) smooth manifold <math>M,</math> and its strong dual space <math>\mathcal{D}^{\prime}(M)</math> of distributions on <math>M,</math>
* the space <math>\mathcal{O}(M)</math> of holomorphic functions on arbitrary complex manifold <math>M,</math> and its strong dual space <math>\mathcal{O}^{\prime}(M)</math> of analytic functionals on <math>M,</math>
* the [[Schwartz space]] <math>\mathcal{S}\left(\R^n\right)</math> on <math>\R^n,</math> and its strong dual space <math>\mathcal{S}^{\prime}\left(\R^n\right)</math> of tempered distributions on <math>\R^n.</math>
</li>
</ol>

=== Counter-examples ===

*There exists a non-reflexive locally convex TVS whose strong dual is reflexive.{{sfn|Schaefer|Wolff|1999|pp=190-202}}