A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces
Volume 172 / 2002
Fundamenta Mathematicae 172 (2002), 117-152
MSC: Primary 46B20.
DOI: 10.4064/fm172-2-3
Abstract
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.
