For a countable ordinal α we denote by Cα the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by α. We show that each Cα admits a separable, reflexive universal space. We also show that spaces in the class C ω α•ω embed into spaces of the same class with a basis. As a consequence we deduce that each Cα is analytic in the Effros-Borel structure of subspaces of C[0, 1].

(College Station, TX), E. Odell (Austin, TX), Th. Schlumprecht (College Station, TX) and A. Zsák (Leeds)Abstract. For every α < ω1 we establish the existence of a separable Banach space whose Szlenk index is ω αω+1 and which is universal for all separable Banach spaces whose Szlenk index does not exceed ω αω . In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with Tsirelson type upper estimates.

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We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given ε > 0, so that the basis becomes (1 + ε)-democratic, and hence (2 + ε)-greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is (1 + ε)-greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applications include the Haar systems in Lp[0, 1], 1 < p < ∞, and in dyadic Hardy space H 1 , as well as the unit vector basis of Tsirelson space.

Let (e i ) be a fundamental system of a Banach space. We consider the problem of approximating linear combinations of elements of this system by linear combinations using quantized coefficients. We will concentrate on systems which are possibly redundant. Our model for this situation will be frames in Banach spaces.