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A Banach space X with a Schauder basis is defined to have the restricted quotient hereditarily indecomposable property if X/Y is hereditarily indecomposable for any infinite-codimensional subspace Y with a successive finite-dimensional decomposition on the basis of X. The following dichotomy theorem is proved: any infinite-dimensional Banach space contains a quotient of a subspace which either has an unconditional basis, or has the restricted quotient hereditarily indecomposable property.

We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following. Let be a Banach space with an unconditional basis ${\left(e_i\right)}_{i\in \mathbb{N}}$; then either there exists a perfect set P of infinite subsets of ℕ such that for any two distinct A,B ∈ P, ${\left[e_i\right]}_{i\in A}\ncong {\left[e_i\right]}_{i\in B}$, or for a residual set of infinite subsets A of ℕ, ${\left[e_i\right]}_{i\in A}$ is isomorphic to , and in that case, is isomorphic to its square, to its hyperplanes, uniformly isomorphic to $\oplus {\left[e_i\right]}_{i\in D}$ for any D ⊂ ℕ, and isomorphic to a denumerable Schauder decomposition...

We analyse several examples of separable Banach spaces, some of them new, and relate them to several dichotomies obtained in [11],by classifying them according to which side of the dichotomies they fall.

We show that the classes of separable reflexive Banach spaces and of spaces with separable dual are strongly bounded. This gives a new proof of a recent result of E. Odell and Th. Schlumprecht, asserting that there exists a separable reflexive Banach space containing isomorphic copies of every separable uniformly convex Banach space.