J an 2 00 2 On the number of non - isomorphic subspaces of a Banach space . Valentin Ferenczi and Christian Rosendal December 2001

We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following: let X be a Banach space with an unconditional basis {ei}i∈N; then either there exists a perfect set P of infinite subsets of N such that for any two distinct A,B ∈ P, [ei]i∈A ≇ [ei]i∈B , or for a residual set of infinite subsets A of N, [ei]i∈A is isomorphic to X, and in that case, X is isomorphic to its square, to ts hyperplanes, uniformly isomorphic to X⊕[ei]i∈D for any D ⊂ N and to a denumerable Schauder decomposition into uniformly isomorphic copies of itself. The starting point of this article is the so called ”homogeneous space problem”, due to S. Banach. It has been solved at the end of last century by W.T. Gowers ([2],[3]), R. KomorowskiN. Tomczak ([7],[8]) and W.T. GowersB. Maurey ([4]); see e.g. [10] for a survey. Recall that a Banach space is said to be homogeneous if it is isomorphic to all its (infinite-dimensional closed) subspaces. The previously named authors showed that l2 is the only homogeneous Banach space. A very natural question was posed to us by G. Godefroy: if a Banach space is not isomorphic to l2, then how many mutually non-isomorphic subspaces must it contain (obviously, at least 2). In this article, we concentrate on spaces with a basis and subspaces of it spanned by subsequences. We shall also be interested in the relation of equivalence of basic sequences. By ”many” we shall mean the Cantor concept of cardinality, and sometimes finer concepts from the theory of classification of equivalence relations in descriptive set theory. 1 Basic notions about basic sequences. Let X be some separable Banach space and {