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

Volume 172 / 2002

Benoit Bossard Fundamenta Mathematicae 172 (2002), 117-152 MSC: Primary 46B20. DOI: 10.4064/fm172-2-3


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.


  • Benoit BossardUniversité Paris 6, Équipe d'Analyse
    Boîte 186, 4 place Jussieu
    75252 Paris Cedex 05, France

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