Editing Reflexive space (section)

== 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>