Szlenk Index Research Articles

Szlenk indices and uniform homeomorphisms

We prove some rather precise renorming theorems for Banach spaces with Szlenk index wo. We use these theorems to show the invariance of certain quantitative Szlenk-type indices under uniform homeomorphisms.

The dual of the bourgain-delbaen space

It is shown that a\(\mathcal{L}_\infty \)-space with separable dual constructed by Bourgain and Delbaen has small Szlenk index and thus does not have a quotient isomorphic toC(ωω). It follows that this is a\(\mathcal{L}_\infty \)-space which is the same size asc 0 in the sense of the Szlenk index but does not containc 0. This has some consequences in the theory of uniform homeomorphism of Banach spaces.

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

Let B be a separable Banach space and let X=B * be separable. We prove that if B has finite Szlenk index (for all e > 0) then B can be renormed to have the weak* uniform Kadec-Klee property. Thus if e > 0 there exists δ (e) > 0 so that if x n is a sequence in the ball of X converging ω* to x so that $$\lim \inf _{n \to \infty } \left\| {x_n - x} \right\| \geqslant \varepsilon {\text{ then }}\left\| x \right\| \leqslant 1 - \delta (\varepsilon )$$ . In addition we show that the norm can be chosen so that δ (e) ≥ cep for some p 0.

L'espace de Banach C0 est déterminé par sa métrique

In this Note we show that a Banach space which is Lipschitz-isomorphic to a subspace of c 0 is linearly isomorphic to a subspace of c 0. We deduce that a space which is Lipschitz-isomorphic to c 0 is in fact linearly isomorphic to c 0. We also prove that a space which is uniformly homeomorphic to a subspace of c 0 has a summable Szlenk index. Finally, we investigate the extension of these results to the non-separable case.

On the Szlenk index and the weak*-dentability index

On the Szlenk index and the weak*-dentability index

Some properties of the Pisier-Zu interpolation spaces

For a closed subset I of the interval [0,1] we let A(I) = [v_1(I),C(I)]_{(1/2)2}. We show that A(I) is isometric to a 1-complemented subspace of A(0,1), and that the Szlenk index of A(I) is larger than the Cantor index of I. We also investigate, for ordin