Banach spaces of bounded Szlenk index

E. Odell¹ ; Th. Schlumprecht² ; A. Zsák³

¹Department of Mathematics The University of Texas 1 University Station C1200 Austin, TX 78712, U.S.A.
²Department of Mathematics Texas A&M University College Station, TX 78712, U.S.A.
³School of Mathematical Sciences University of Nottingham University Park Nottingham NG7 2RD, United Kingdom

Studia Mathematica, Tome 183 (2007) no. 1, pp. 63-97

Résumé

For a countable ordinal $\alpha$ we denote by ${\cal C}_\alpha$ the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by $\alpha$. We show that each ${\cal C}_\alpha$ admits a separable, reflexive universal space. We also show that spaces in the class ${\cal C}_{\omega^{\alpha\cdot\omega}}$ embed into spaces of the same class with a basis. As a consequence we deduce that each ${\cal C}_\alpha$ is analytic in the Effros–Borel structure of subspaces of $C[0,1]$.

Détail

DOI : 10.4064/sm183-1-4

Keywords: countable ordinal alpha denote cal alpha class separable reflexive banach spaces whose szlenk index szlenk index their dual bounded alpha each cal alpha admits separable reflexive universal space spaces class cal omega alpha cdot omega embed spaces class basis consequence deduce each cal alpha analytic effros borel structure subspaces