Editing Reflexive space (section)

=== 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" />