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When the set of closed subspaces of C(∆), where ∆ 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 `1(ω), . . .) 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. 0. Introduction. Classifying Banach spaces is notoriously difficult, and it is natural to conjecture that the obstruction to such a classification lies in the topological complexity of the relevant relations, and in particular of the isomorphism equivalence relation. To support this conjecture, one needs of course a natural and usable frame in which such topological notions can be handled. The purpose of the present work is to provide such a frame. The collection of separable Banach spaces is not a set, and we first need a proper parametrization of this collection. We choose to consider it as the set of all closed subspaces of the space C(∆) of continuous functions on the Cantor set. It is indeed well known that every separable Banach space is isometric to a subspace of C(∆). This choice could be considered as arbitrary; however we show that natural but different choices lead to the same levels of complexity. To investigate the topological complexity of natural families of Banach spaces, we use the theory of analytic sets, introduced by Suslin and Lusin. The basic results of this theory are presented e.g. in [K-L1], [K] or [Z]. There is a strong interplay between analytic sets and classical analysis, for which 2000 Mathematics Subject Classification: Primary 46B20.

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