Th. Schlumprecht

On Asymptotic Structure, the Szlenk Index and UKK Properties in Banach Spaces

AbstractLet B be a separable Banach space and let X=B* be separable. We prove that if B has finite Szlenk index (for all ε > 0) then B can be renormed to have the weak* uniform Kadec-Klee property. Thus if ε > 0 there exists δ (ε) > 0 so that if xn is a sequence in the ball of X converging ω* to x so that

Coefficient quantization for frames in Banach spaces

\lim \inf _{n \to \infty } \left\| {x_n - x} \right\| \geqslant \varepsilon {\text{ then }}\left\| x \right\| \leqslant 1 - \delta (\varepsilon )

Distortion and Stabilized Structure in Banach Spaces; New Geometric Phenomena for Banach and Hilbert Spaces

. In addition we show that the norm can be chosen so that δ (ε) ≥ cεp for some p < ∞ and c >0.

EQUILATERAL SETS IN UNIFORMLY SMOOTH BANACH SPACES

We give examples of two Banach spaces. One Banach space has no spreading model which contains l p (1 ≤ p < ∞) or c o. The other space has an unconditional basis for which l p (1 ≥ p < ∞) and c o are block finitely represented in all block bases.

A universal reflexive space for the class of uniformly convex Banach spaces

Let (ei) 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.

Embedding uniformly convex spaces into spaces with very few operators

The unit sphere of Hilbert space, ℓ2, is shown to contain a remarkable sequence of nearly orthogonal sets. Precisely, there exist a sequence of sets of norm one elements of ℓ2, (Ci)i=1∞, and reals εi↓0 so that a) each setCi has nonempty intersection with every infinite dimensional closed subspace of ℓ2 and b) fori≠j,x∈C, andy∈Cj, |〈x, y〉|

ON THE STRUCTURE OF ASYMPTOTIC p SPACES

Many of the fundamental research problems in the geometry of normed linear spaces can be loosely phrased as: Given a Banach space X and a class of Banach spaces Y does X contain a subspace Y ∈ Y? As a Banach space X is determined by its unit ball B x ≡ { x ∈ X :‖ x ‖ ≤ 1 } the problem can be rephrased in terms of the geometry of convex sets: Can a given unit ball B x be sliced with a subspace to obtain a set in some given class of unit balls? A result of this type is the famous theorem of Dvoretzky (see also [L], [M6], [M4], [MS], [FLM]).

Greedy Bases for Besov Spaces

Let