Editing Reflexive space

{{Short description|Locally convex topological vector space}}
In the area of mathematics known as [[functional analysis]], a '''reflexive space''' is a [[locally convex]] [[topological vector space]] for which the canonical evaluation map from <math>X</math> into its [[bidual]] (which is the [[strong dual]] of the strong dual of <math>X</math>) is a [[homeomorphism]] (or equivalently, a [[TVS isomorphism]]). 
A [[normed space]] is reflexive if and only if this canonical evaluation map is [[surjective]], in which case this (always linear)  evaluation map is an [[isometric isomorphism]] and the normed space is a [[Banach space]]. Those spaces for which the canonical evaluation map is surjective are called [[semi-reflexive]] spaces. 

In 1951, [[Robert C. James|R. C. James]] discovered a Banach space, now known as [[James' space]], that is {{em|not}} reflexive (meaning that the canonical evaluation map is not an isomorphism) but is nevertheless isometrically isomorphic to its bidual (any such [[isometric isomorphism]] is necessarily {{em|not}} the canonical evaluation map). So importantly, for a Banach space to be reflexive, it is not enough for it to be isometrically isomorphic to its bidual; it is the canonical evaluation map in particular that has to be a homeomorphism.

Reflexive spaces play an important role in the general theory of [[locally convex]] TVSs and in the theory of [[Banach space]]s in particular.  [[Hilbert space]]s are prominent examples of reflexive Banach spaces.  Reflexive Banach spaces are often characterized by their geometric properties.

== Definition ==

;Definition of the bidual

{{Main|Bidual}}

Suppose that <math>X</math> 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 <math>X</math> (that is, for any <math>x \in X, x \neq 0</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> (some texts write <math>X^{\prime}_\beta</math>) denote the [[strong dual]] of <math>X,</math> which is the vector space <math>X^{\prime}</math> of continuous linear functionals on <math>X</math> endowed with the [[topology of uniform convergence]] on [[Topology of uniform convergence#Strong dual topology b(X*, X)|bounded subsets]] of <math>X</math>; 
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 <math>X</math> is a normed space, then the strong dual of <math>X</math> is the continuous dual space <math>X^{\prime}</math> with its usual norm topology. 
The '''bidual''' of <math>X,</math> 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}} 
If <math>X</math> is a normed space, then <math>X^{\prime\prime}</math> is the continuous dual space of the Banach space <math>X^{\prime}_b</math> with its usual norm topology. 

;Definitions of the evaluation map and reflexive spaces

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 a linear map called the '''evaluation map at <math>x</math>'''; 
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 <math>X,</math> the linear 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'''. 
Call <math>X</math> '''[[semi-reflexive]]''' if <math>J : X \to \left(X^{\prime}_b\right)^{\prime}</math> is bijective (or equivalently, [[surjective]]) and we call <math>X</math> '''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}}
A [[normable]] space is reflexive if and only if it is semi-reflexive or equivalently, if and only if the evaluation map is surjective.

== Reflexive Banach spaces ==

Suppose <math>X</math> is a [[normed vector space]] over the number field <math>\mathbb{F} = \R</math> or <math>\mathbb{F} = \Complex</math> (the [[real number]]s or the [[complex number]]s), with a norm <math>\|\,\cdot\,\|.</math> Consider its [[dual norm|dual normed space]] <math>X^{\prime},</math> that consists of all [[continuous function|continuous]] [[linear functional]]s <math>f : X \to \mathbb{F}</math> and is equipped with the [[dual norm]] <math>\|\,\cdot\,\|^{\prime}</math> defined by
<math display="block">\|f\|^{\prime} = \sup \{ |f(x)| \,:\, x \in X, \ \|x\| = 1 \}.</math>

The dual <math>X^{\prime}</math> is a normed space (a [[Banach space]] to be precise), and its dual normed space <math>X^{\prime\prime} = \left(X^{\prime}\right)^{\prime}</math> is called '''bidual space''' for <math>X.</math> The bidual consists of all continuous linear functionals <math>h : X^{\prime}\to \mathbb{F}</math> and is equipped with the norm <math>\|\,\cdot\,\|^{\prime\prime}</math> dual to <math>\|\,\cdot\,\|^{\prime}.</math> Each vector <math>x \in X</math> generates a scalar function <math>J(x) : X^{\prime} \to \mathbb{F}</math> by the formula:
<math display="block">J(x)(f) = f(x) \qquad \text{ for all } f \in X^{\prime},</math>
and <math>J(x)</math> is a continuous linear functional on <math>X^{\prime},</math> that is, <math>J(x)\in X^{\prime\prime}.</math>  One obtains in this way a map
<math display="block">J : X \to X^{\prime\prime}</math>
called '''evaluation map''', that is linear. It follows from the [[Hahn–Banach theorem]] that <math>J</math> is injective and preserves norms:
<math display="block">\text{ for all } x \in X \qquad \|J(x)\|^{\prime\prime} = \|x\|,</math>
that is, <math>J</math> maps <math>X</math> isometrically onto its image <math>J(X)</math> in <math>X^{\prime\prime}.</math>  Furthermore, the image <math>J(X)</math> is closed in <math>X^{\prime\prime},</math> but it need not be equal to <math>X^{\prime\prime}.</math>

A normed space <math>X</math> is called '''reflexive''' if it satisfies the following equivalent conditions:
<ol type="i">
<li> the evaluation map <math>J : X \to X^{\prime\prime}</math> is [[Bijection, injection and surjection|surjective]],</li>
<li> the evaluation map <math>J : X \to X^{\prime\prime}</math> is an [[Banach space|isometric isomorphism]] of normed spaces,</li>
<li> the evaluation map <math>J : X \to X^{\prime\prime}</math> is an [[Banach space|isomorphism]] of normed spaces.</li>
</ol>
A reflexive space <math>X</math> is a Banach space, since <math>X</math> is then isometric to the Banach space <math>X^{\prime\prime}.</math>

=== Remark ===

A Banach space <math>X</math> is reflexive if it is linearly isometric to its bidual under this canonical embedding <math>J.</math> [[James' space]] is an example of a non-reflexive space which is linearly isometric to its [[Dual space#Double dual|bidual]]. Furthermore, the image of James' space under the canonical embedding <math>J</math> has [[codimension]] one in its bidual.
<ref>{{cite journal|author=Robert C. James |author-link=Robert C. James |title=A non-reflexive Banach space isometric with its second conjugate space|journal=Proc. Natl. Acad. Sci. U.S.A.|volume=37|pages=174–177|year=1951|issue=3|doi=10.1073/pnas.37.3.174|pmc=1063327|pmid=16588998|bibcode=1951PNAS...37..174J|doi-access=free}}</ref>
A Banach space <math>X</math> is called '''quasi-reflexive''' (of order <math>d</math>) if the quotient <math>X^{\prime\prime} / J(X)</math> has finite dimension <math>d.</math>

=== Examples ===

# Every finite-dimensional normed space is reflexive, simply because in this case, the space, its dual and bidual all have the same linear dimension, hence the linear injection <math>J</math> from the definition is bijective, by the [[rank–nullity theorem]].
# The Banach space [[Sequence space#c and c0|<math>c_0</math>]] of scalar sequences tending to 0 at infinity, equipped with the supremum norm, is not reflexive.  It follows from the general properties below that [[Sequence space#.E2.84.93p spaces|<math>\ell^1</math> and <math>\ell^{\infty}</math>]] are not reflexive, because <math>\ell^1</math> is isomorphic to the dual of <math>c_0</math> and <math>\ell^{\infty}</math> is isomorphic to the dual of <math>\ell^1.</math>
# All [[Hilbert space]]s are reflexive, as are the [[Lp space]]s <math>L^p</math> for <math>1 < p < \infty.</math> More generally: all [[uniformly convex space|uniformly convex]] Banach spaces are reflexive according to the [[Milman–Pettis theorem]].  The <math>L^1(\mu)</math> and <math>L^{\infty}(\mu)</math> spaces are not reflexive (unless they are finite dimensional, which happens for example when <math>\mu</math> is a measure on a finite set).  Likewise, the Banach space <math>C([0, 1])</math> of continuous functions on <math>[0, 1]</math> is not reflexive.
# The spaces <math>S_p(H)</math> of operators in the [[Schatten class operator|Schatten class]]  on a Hilbert space <math>H</math> are uniformly convex, hence reflexive, when <math>1 < p < \infty.</math>  When the dimension of <math>H</math> is infinite, then <math>S_1(H)</math> (the [[trace class]]) is not reflexive, because it contains a subspace isomorphic to <math>\ell^1,</math> and <math>S_{\infty}(H) = L(H)</math> (the bounded linear operators on <math>H</math>) is not reflexive, because it contains a subspace isomorphic to <math>\ell^{\infty}.</math>  In both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of <math>H.</math>

=== Properties ===

Since every finite-dimensional normed space is a reflexive [[Banach space]], only infinite-dimensional spaces can be non-reflexive.

If a Banach space <math>Y</math> is isomorphic to a reflexive Banach space <math>X</math> then <math>Y</math> is reflexive.<ref>Proposition 1.11.8 in {{harvtxt|Megginson|1998|p=99}}.</ref>

Every [[Closed set|closed]] [[linear subspace]] of a reflexive space is reflexive.  The continuous dual of a reflexive space is reflexive.  Every [[Quotient space (linear algebra)|quotient]] of a reflexive space by a closed subspace is reflexive.<ref>{{harvtxt|Megginson|1998|pp=104&ndash;105}}.</ref>

Let <math>X</math> be a Banach space. The following are equivalent.
<ol>
<li>The space <math>X</math> is reflexive.</li>
<li>The continuous dual of <math>X</math> is reflexive.<ref>Corollary 1.11.17, p.&nbsp;104 in {{harvtxt|Megginson|1998}}.</ref></li>
<li>The closed unit ball of <math>X</math> is [[compact space|compact]] in the [[weak topology]]. (This is known as Kakutani's Theorem.){{sfn|Conway|1985|loc=Theorem V.4.2, p.&nbsp;135}}</li>
<li>Every bounded sequence in <math>X</math> has a weakly convergent subsequence.<ref>Since weak compactness and weak sequential compactness coincide by the [[Eberlein–Šmulian theorem]].</ref></li>
<li>The statement of [[Riesz's lemma]] holds when the real number<ref group=note>The statement of [[Riesz's lemma]] involves only one real number, which is denoted by <math>\alpha</math> in the article on Riesz's lemma. The lemma always holds for all real <math>\alpha < 1.</math> But for a Banach space, the lemma holds for all <math>\alpha \leq 1</math> if and only if the space is reflexive.</ref> is exactly <math>1.</math>{{sfn|Diestel|1984|p=6}} Explicitly, for every closed proper vector subspace <math>Y</math> of <math>X,</math> there exists some vector <math>u \in X</math> of unit norm <math>\|u\| = 1</math> such that <math>\|u - y\| \geq 1</math> for all <math>y \in Y.</math> 
* Using <math>d(u, Y) := \inf_{y \in Y} \|u - y\|</math> to denote the distance between the vector <math>u</math> and the set <math>Y,</math> this can be restated in simpler language as: <math>X</math> is reflexive if and only if for every closed proper vector subspace <math>Y,</math> there is some vector <math>u</math> on the [[unit sphere]] of <math>X</math> that is always at least a distance of <math>1 = d(u, Y)</math> away from the subspace. 
* For example, if the reflexive Banach space <math>X = \Reals^3</math> is endowed with the usual [[Euclidean norm]] and <math>Y = \Reals \times \Reals \times \{0\}</math> is the <math>x-y</math> plane then the points <math>u = (0, 0, \pm 1)</math> satisfy the conclusion <math>d(u, Y) = 1.</math> If <math>Y</math> is instead the <math>z</math>-axis then every point belonging to the unit circle in the <math>x-y</math> plane satisfies the conclusion. 
</li>
<li>Every continuous linear functional on <math>X</math> attains its supremum on the closed unit ball in <math>X.</math><ref>Theorem 1.13.11 in {{harvtxt|Megginson|1998|p=125}}.</ref> ([[James' theorem]])</li>
</ol>

Since norm-closed [[Convex set|convex subsets]] in a Banach space are weakly closed,<ref>Theorem 2.5.16 in {{harvtxt|Megginson|1998|p=216}}.</ref> 
it follows from the third property that closed bounded convex subsets of a reflexive space <math>X</math> are weakly compact.  Thus, for every decreasing sequence of non-empty closed bounded convex subsets of <math>X,</math> the intersection is non-empty.  As a consequence, every continuous [[convex function]] <math>f</math> on a closed convex subset <math>C</math> of <math>X,</math> such that the set
<math display="block">C_t = \{ x \in C \,:\, f(x) \leq t \}</math>
is non-empty and bounded for some real number <math>t,</math> attains its minimum value on <math>C.</math>

The promised geometric property of reflexive Banach spaces is the following: if <math>C</math> is a closed non-empty [[Convex set|convex]] subset of the reflexive space <math>X,</math> then for every <math>x \in X</math> there exists a <math>c \in C</math> such that <math>\|x - c\|</math> minimizes the distance between <math>x</math> and points of <math>C.</math>  This follows from the preceding result for convex functions, applied to<math>f(y) + \|y - x\|.</math>  Note that while the minimal distance between <math>x</math> and <math>C</math> is uniquely defined by <math>x,</math> the point <math>c</math> is not.  The closest point <math>c</math> is unique when <math>X</math> is uniformly convex.

A reflexive Banach space is [[Separable space|separable]] if and only if its continuous dual is separable.  This follows from the fact that for every normed space <math>Y,</math> separability of the continuous dual <math>Y^{\prime}</math> implies separability of <math>Y.</math><ref>Theorem&nbsp;1.12.11 and Corollary&nbsp;1.12.12 in {{harvtxt|Megginson|1998|pp=112–113}}.</ref>

=== Super-reflexive space ===

Informally, a super-reflexive Banach space <math>X</math> has the following property: given an arbitrary Banach space <math>Y,</math> if all finite-dimensional subspaces of <math>Y</math> have a very similar copy sitting somewhere in <math>X,</math> then <math>Y</math> must be reflexive.  By this definition, the space <math>X</math> itself must be reflexive.  As an elementary example, every Banach space <math>Y</math> whose two dimensional subspaces are [[Isometry|isometric]] to subspaces of <math>X = \ell^2</math> satisfies the [[parallelogram law]], hence<ref>see this [[Banach space#Characterizations of Hilbert space among Banach spaces|characterization of Hilbert space among Banach spaces]]</ref> 
<math>Y</math> is a Hilbert space, therefore <math>Y</math> is reflexive.  So <math>\ell^2</math> is super-reflexive.

The formal definition does not use isometries, but almost isometries.  A Banach space <math>Y</math> is '''finitely representable'''<ref name="SRBS">James, Robert C. (1972), "Super-reflexive Banach spaces", Can. J. Math. '''24''':896&ndash;904.</ref> 
in a Banach space <math>X</math> if for every finite-dimensional subspace <math>Y_0</math> of <math>Y</math> and every <math>\epsilon > 0,</math> there is a subspace <math>X_0</math> of <math>X</math> such that the multiplicative [[Banach–Mazur compactum|Banach&ndash;Mazur distance]] between <math>X_0</math> and <math>Y_0</math> satisfies
<math display="block">d\left(X_0, Y_0\right) < 1 + \varepsilon.</math>

A Banach space finitely representable in <math>\ell^2</math> is a Hilbert space.  Every Banach space is finitely representable in <math>c_0.</math>  The [[Lp space]] <math>L^p([0, 1])</math> is finitely representable in <math>\ell^p.</math>

A Banach space <math>X</math> is '''super-reflexive''' if all Banach spaces <math>Y</math> finitely representable in <math>X</math> are reflexive, or, in other words, if no non-reflexive space <math>Y</math> is finitely representable in <math>X.</math>  The notion of [[ultraproduct]] of a family of Banach spaces<ref>Dacunha-Castelle, Didier; Krivine, Jean-Louis (1972), "Applications des ultraproduits à l'étude des espaces et des algèbres de Banach" (in French), Studia Math. '''41''':315&ndash;334.</ref> 
allows for a concise definition: the Banach space <math>X</math> is super-reflexive when its ultrapowers are reflexive.

James proved that a space is super-reflexive if and only if its dual is super-reflexive.<ref name="SRBS" />

=== Finite trees in Banach spaces ===

One of James' characterizations of super-reflexivity uses the growth of separated trees.<ref name="Tree">see {{harvtxt|James|1972}}.</ref>
The description of a vectorial binary tree begins with a [[rooted binary tree]] labeled by vectors: a tree of [[Tree (graph theory)#Rooted tree|height]] <math>n</math> in a Banach space <math>X</math> is a family of <math>2^{n+1} - 1</math> vectors of <math>X,</math> that can be organized in successive levels, starting with level&nbsp;0 that consists of a single vector <math>x_{\varnothing},</math> the [[Tree (graph theory)#Definitions|root]] of the tree, followed, for <math>k = 1, \ldots, n,</math> by a family of <math>s^k</math>2 vectors forming level <math>k:</math>
<math display="block">\left\{ x_{\varepsilon_1, \ldots, \varepsilon_k} \right\}, \quad \varepsilon_j = \pm 1, \quad j = 1, \ldots, k,</math>
that are the [[Tree (graph theory)#Definitions|children]] of vertices of level <math>k - 1.</math>  In addition to the [[Tree (graph theory)|tree structure]], it is required here that each vector that is an [[Tree (graph theory)#Definitions|internal vertex]] of the tree be the midpoint between its two children:
<math display="block">x_\emptyset = \frac{x_1 + x_{-1}}{2}, \quad x_{\varepsilon_1, \ldots, \varepsilon_k} = \frac{x_{\varepsilon_1, \ldots, \varepsilon_k, 1} + x_{\varepsilon_1, \ldots, \varepsilon_k, -1}} {2}, \quad 1 \leq k < n.</math>

Given a positive real number <math>t,</math> the tree is said to be '''<math>t</math>-separated''' if for every internal vertex, the two children are <math>t</math>-separated in the given space norm:
<math display="block">\left\|x_1 - x_{-1}\right\| \geq t, \quad \left\|x_{\varepsilon_1, \ldots, \varepsilon_k, 1} - x_{\varepsilon_1, \ldots, \varepsilon_k, -1}\right\| \geq t, \quad 1 \leq k < n.</math>

<blockquote>'''Theorem.'''<ref name="Tree" />
The Banach space <math>X</math> is super-reflexive if and only if for every <math>t \in (0, 2 \pi],</math> there is a number <math>n(t)</math> such that every <math>t</math>-separated tree contained in the unit ball of <math>X</math> has height less than <math>n(t).</math></blockquote>

[[Uniformly convex space]]s are super-reflexive.<ref name="Tree" />
Let <math>X</math> be uniformly convex, with [[Modulus and characteristic of convexity|modulus of convexity]] <math>\delta_X</math> and let <math>t</math> be a real number in <math>(0, 2].</math>  By the [[Modulus and characteristic of convexity#Definitions|properties]] of the modulus of convexity, a <math>t</math>-separated tree of height <math>n,</math> contained in the unit ball, must have all points of level <math>n - 1</math> contained in the ball of radius <math>1 - \delta_X(t) < 1.</math> By induction, it follows that all points of level <math>n - k</math> are contained in the ball of radius
<math display="block">\left(1 - \delta_X(t)\right)^j, \ j = 1, \ldots, n.</math>

If the height <math>n</math> was so large that 
<math display="block">\left(1 - \delta_X(t)\right)^{n-1} < t / 2,</math>
then the two points <math>x_1, x_{-1}</math> of the first level could not be <math>t</math>-separated, contrary to the assumption.  This gives the required bound <math>n(t),</math> function of <math>\delta_X(t)</math> only.

Using the tree-characterization, [[Per Enflo|Enflo]] proved<ref>{{cite journal
| last1=Enflo | first1=Per | authorlink1=Per Enflo
| date=1972
| title=Banach spaces which can be given an equivalent uniformly convex norm
| journal=[[Israel Journal of Mathematics]]
| volume=13
| issue=3–4 | pages=281&ndash;288
| doi=10.1007/BF02762802 | doi-access=}}</ref> 
that super-reflexive Banach spaces admit an equivalent uniformly convex norm.  Trees in a Banach space are a special instance of vector-valued [[Martingale (probability theory)|martingales]].  Adding techniques from scalar martingale theory, [[Gilles Pisier|Pisier]] improved Enflo's result by showing<ref>{{cite journal
| last1=Pisier | first1=Gilles | authorlink1=Gilles Pisier
| date=1975
| title=Martingales with values in uniformly convex spaces
| journal=[[Israel Journal of Mathematics]]
| volume=20
| issue=3–4 | pages=326&ndash;350
| doi=10.1007/BF02760337 | doi-access=}}</ref> that a super-reflexive space <math>X</math> admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant <math>c > 0</math> and some real number <math>q \geq 2,</math>
<math display="block">\delta_X(t) \geq c \, t^q, \quad \text{ whenever } t \in [0, 2].</math>

== 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}}

== Other types of reflexivity ==

A stereotype space, or polar reflexive space, is defined as a [[topological vector space]] (TVS) satisfying a similar condition of reflexivity, but with the topology of uniform convergence on [[Totally bounded set|totally bounded]] subsets (instead of [[Bounded set|bounded]] subsets) in the definition of dual space <math>X^{\prime}.</math> More precisely, a TVS <math>X</math> is called polar reflexive<ref>{{cite book|last=Köthe|first=Gottfried|title=Topological Vector Spaces I|publisher=Springer|series=Springer Grundlehren der mathematischen Wissenschaften|year=1983|url=https://www.springer.com/gp/book/9783642649905#aboutBook|isbn=978-3-642-64988-2 }}</ref> or stereotype if the evaluation map into the second dual space
<math display="block">J : X \to X^{\star\star},\quad J(x)(f) = f(x),\quad x\in X,\quad f\in X^\star</math>
is an [[TVS-isomorphism|isomorphism of topological vector spaces]].<ref name=isomorphism /> Here the stereotype dual space <math>X^\star</math> is defined as the space of continuous linear functionals <math>X^{\prime}</math> endowed with the topology of uniform convergence on totally bounded sets in <math>X</math> (and the ''stereotype second dual space'' <math>X^{\star\star}</math> is the space dual to <math>X^{\star}</math> in the same sense).

In contrast to the classical reflexive spaces the class '''Ste''' of stereotype spaces is very wide (it contains, in particular, all [[Fréchet space]]s and thus, all [[Banach space]]s), it forms a [[closed monoidal category]], and it admits standard operations (defined inside of '''Ste''') of constructing new spaces, like taking closed subspaces, quotient spaces, projective and injective limits, the space of operators, tensor products, etc. The category '''Ste''' have applications in duality theory for non-commutative groups.

Similarly, one can replace the class of bounded (and totally bounded) subsets in <math>X</math> in the definition of dual space <math>X^{\prime},</math> by other classes of subsets, for example, by the class of compact subsets in <math>X</math> – the spaces defined by the corresponding reflexivity condition are called {{em|reflective}},<ref>{{cite journal|last=Garibay Bonales|first=F.|author2=Trigos-Arrieta, F. J.|author3=Vera Mendoza, R.|title=A characterization of Pontryagin-van Kampen duality for locally convex spaces|journal=Topology and Its Applications|year=2002|volume=121|issue=1–2 |pages=75–89|doi=10.1016/s0166-8641(01)00111-0|doi-access=free}}</ref><ref>{{cite journal|last=Akbarov|first=S. S.|author2=Shavgulidze, E. T.|title=On two classes of spaces reflexive in the sense of Pontryagin|journal=Mat. Sbornik|year=2003|volume=194|issue=10|pages=3–26}}</ref> and they form an even wider class than '''Ste''', but it is not clear (2012), whether this class forms a category with properties similar to those of '''Ste'''.

== See also ==

* {{annotated link|Grothendieck space}}
** A generalization which has some of the properties of reflexive spaces and includes many spaces of practical importance is the concept of [[Grothendieck space]].
* {{annotated link|Reflexive operator algebra}}

==References==

=== Notes ===

{{reflist|group=note}}

===Citations===

{{reflist}}

===General references===

* {{citation|last=Bernardes|first=Nilson C. Jr.|title=On nested sequences of convex sets in Banach spaces|pages=558&ndash;561|volume=389|publisher=Journal of Mathematical Analysis and Applications|year=2012}} <!-- {{sfn|Bernardes|2012|p=}} -->.
* {{cite book|author-link=John B. Conway|first=John B.|last=Conway|title=A Course in Functional Analysis|publisher=Springer|year=1985}}
* {{cite book|last=Diestel|first=Joe|title=Sequences and series in Banach spaces|publisher=Springer-Verlag|publication-place=New York|date=1984|isbn=0-387-90859-5|oclc=9556781}} <!--{{sfn|Diestel|1984|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}}
* {{citation
|last=James
|first=Robert C.
|title=Some self-dual properties of normed linear spaces. Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967)
|pages=159&ndash;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}}
* {{citation
|last=Megginson
|first=Robert E.|author-link=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=}} -->
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} --> 
* {{Cite book|last=Schaefer|first=Helmut H.|author-link=Helmut H. Schaefer|year=1966|title=Topological vector spaces|publisher=The Macmillan Company|location=New York}}
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->

{{Banach spaces}}
{{Functional analysis}}
{{Topological vector spaces}}

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