A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces

Benoit Bossard¹

¹ Université Paris 6, Équipe d'Analyse Boîte 186, 4 place Jussieu 75252 Paris Cedex 05, France

Fundamenta Mathematicae, Tome 172 (2002) no. 2, pp. 117-152

Résumé

When the set of closed subspaces of $C({\mit \Delta })$, where ${\mit \Delta }$ is the Cantor set, is equipped with the standard Effros–Borel structure, the graph of the basic relations between Banach spaces (isomorphism, being isomorphic to a subspace, quotient, direct sum, …) is analytic non-Borel. Many natural families of Banach spaces (such as reflexive spaces, spaces not containing $\ell _1(\omega ),\dots$) are coanalytic non-Borel. Some natural ranks (rank of embedding, Szlenk indices) are shown to be coanalytic ranks. Applications are given to universality questions. Analogous results are shown for basic sequences modulo equivalence.

Détail

DOI : 10.4064/fm172-2-3

Keywords: set closed subspaces mit delta where mit delta cantor set equipped standard effros borel structure graph basic relations between banach spaces isomorphism being isomorphic subspace quotient direct sum hellip analytic non borel many natural families banach spaces reflexive spaces spaces containing ell omega dots coanalytic non borel natural ranks rank embedding szlenk indices shown coanalytic ranks applications given universality questions analogous results shown basic sequences modulo equivalence