We characterize those classes C of separable Banach spaces admitting a separable universal space Y (that is, a space Y containing, up to isomorphism, all members of C) which is not universal for all separable Banach spaces. The characterization is a byproduct of the fact, proved in the paper, that the class NU of non-universal separable Banach spaces is strongly bounded. This settles in the aff...

We show that if a separable Banach space Z contains isometric copies of every strictly convex separable Banach space, then Z actually contains an isometric copy of every separable Banach space. We prove that if Y is any separable Banach space of dimension at least 2, then the collection of separable Banach spaces which contain an isometric copy of Y is analytic non Borel.

For every α < ω1 we establish the existence of a separable Banach space whose Szlenk index is ω and which is universal for all separable Banach spaces whose Szlenkindex does not exceed ω. In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with upper estimates.

We investigate various kinds of bases in infinite dimensional Banach spaces. In particular, we consider the complexity of Hamel bases in separable and non-separable Banach spaces and show that in a separable Banach space a Hamel basis cannot be analytic, whereas there are non-separable Hilbert spaces which have a discrete and closed Hamel basis. Further we investigate the existence of certain c...

We show that there exists a separable reflexive Banach space into which every separable uniformly convex Banach space isomorphically embeds. This solves a problem of J. Bourgain. We also give intrinsic characterizations of separable reflexive Banach spaces which embed into a reflexive space with a block q-Hilbertian and/or a block p-Besselian finite dimensional decomposition.

1. Introduction and notation. In this paper the word " local " is used in at least three different meanings. Our aim is to study local Banach spaces of Fréchet or other locally convex spaces, and it turns out that it is convenient to use the local theory of Banach spaces for this purpose. Recall that given a locally convex space E and a continuous seminorm p on E the completion of the normed sp...

We prove a structural property of the class of unconditionally saturated separable Banach spaces. We show, in particular, that for every analytic set A, in the Effros-Borel space of subspaces of C[0, 1], of unconditionally saturated separable Banach spaces, there exists an unconditionally saturated Banach space Y , with a Schauder basis, that contains isomorphic copies of every space X in the c...

We characterize those classes C of separable Banach spaces for which there exists a separable Banach space Y not containing l1 and such that every space in the class C is a quotient of Y .

Table of Contents 1. Introduction. 2. The isomorphic classification of separable C(K) spaces. A. Milutin's Theorem. B. C(K) spaces with separable dual via the Szlenk index 3. Some Banach space properties of separable C(K) spaces. A. Weak injectivity. B. c 0-saturation of spaces with separable dual. C. Uncomplemented embeddings of C([0, 1]) and C(ω ω +) in themselves. 4. Operators on C(K)-spaces.