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Banach spaces of bounded Szlenk index

E. Odell;  Th. Schlumprecht;  A. Zsák — 2007

Studia Mathematica

For a countable ordinal α we denote by α the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by α. We show that each α admits a separable, reflexive universal space. We also show that spaces in the class ω α · ω embed into spaces of the same class with a basis. As a consequence we deduce that each α is analytic in the Effros-Borel structure of subspaces of C[0,1].

Banach spaces of bounded Szlenk index II

D. Freeman;  E. Odell;  Th. Schlumprecht;  A. Zsák — 2009

Fundamenta Mathematicae

For every α < ω₁ we establish the existence of a separable Banach space whose Szlenk index is ω α ω + 1 and which is universal for all separable Banach spaces whose Szlenk index does not exceed ω α ω . In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with Tsirelson type upper estimates.

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