Editing Reflexive space (section)

=== 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>