Prescribed Szlenk index of separable Banach spaces

Abstract

In a previous work, the first named author described the set P of all values of the Szlenk indices of separable Banach spaces. We complete this result by showing that for any integer n and any ordinal α in P, there exists a separable Banach space X such that the Szlenk of the dual of order k of X is equal to the first infinite ordinal ω for all k in {0, .., n − 1} and equal to α for k = n. One of the ingredients is to show that the Lindenstrauss space and its dual both have a Szlenk index equal to ω. We also show that any element of P can be realized as a Szlenk index of a reflexive Banach space with an unconditional basis.

On the Coarse Geometry of James Spaces

“…For the first step of the proof we shall exploit the existence of an equivalent q-AUS norm | | on X * (we also denote | | its dual norm on X * * ). It is worth mentioning that if X is not reflexive, | | cannot be the dual norm of an equivalent norm on X (see for instance Proposition 2.6 in [7]). Assume also that there exists b > 0 such that…”

On Asymptotic Uniform Smoothness and Nonlinear Geometry of Banach Spaces

“…Those results were actually only proven for p = p = 2 in[25] and[13], but a simple adaptation of their proof gives us this general result.…”

“…In particular, Z p,X and Z * p,X are both separable. Furthermore, it was proven in Theorem 2.1 of [CL17] that Z p,X is p-AUSable and that Z * p,X is p ′ -AUSable, where p ′ is the conjugate of p, i.e., 1/p + 1/p ′ = 1. 9 In particular, if X is infinite dimensional, those spaces are not quasireflexive.…”

On the weak maximizing properties

“…It is worth mentioning that if X ≡ Z * and the dual norm • Z * is AUS, then X must be reflexive (see for instance Proposition 2.6 in [9]). The following proposition is elementary.…”