Banach spaces of bounded Szlenk index II

Abstract

(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.

Power type asymptotically uniformly smooth and asymptotically uniformly flat norms

“…However, that proof uses the separability in a fundamental way and cannot be modified to work for the non-separable case. Furthermore, [11] deduces the content of Theorem 1.1 for separable spaces from a separate, quite involved result. Our proof is short and direct, and is in the spirit of Pisier's famous renorming theorem [21].…”

“…We will be concerned with the case E = c 0 or E = ℓ p for some 1 < p < ∞. We note that for Banach spaces with separable dual, the notion of ℓ p or c 0 upper tree estimates has already been defined in the literature (see, for example, [11]). Our presentation of the definition differs from what is commonly given in the literature, but we discuss in Section 3 how our presentation differs from the usual one, but the underlying property coincides with the usual one.…”

“…Applying Theorem 1.1 to the identity operator of a Banach space yields new renorming theorems for non-separable Banach spaces. In the case that A = I X and X is separable, Theorem 1.1 follows from [11]. However, that proof uses the separability in a fundamental way and cannot be modified to work for the non-separable case.…”

The universality of ℓ 1 as a dual space

“…So, in proving Theorem A, we could begin with such a space. However, to make our construction work, we need a quantified version of this theorem which appears in [12]. For Theorem C, we need a quantified reflexive version [26].…”

“…If S z (X ) < ω 1 then S z (X ) = ω β for some β < ω 1 . Much has been written on the Szlenk index (e.g., see [3,6,[12][13][14]20,21,26]). We note that the upper and lower estimates in both theorems are with respect to the unit vector basis (t i ) of T c,α and its biorthogonal sequence (t * i ), a basis for T * c,α .…”