Three and a half asymptotic properties

Abstract

We define and discuss transfinite asymptotic notions of smoothability, type, and equal norm type. We prove distinctness of these notions for a proper class of ordinals and that each class is an ideal. We also extend some results from a 2001 paper by Godefroy, Kalton, and Lancien to operators and ordinals greater than zero regarding the equivalence of equal norm asymptotic type and uniform renormings with power type smoothness. Finally, we discuss an extension of a non-linear result for quasireflexive, asymptotically p-smoothable Banach spaces to quasi-reflexive Banach spaces with asymptotic equal norm type p.

On Asymptotic Uniform Smoothness and Nonlinear Geometry of Banach Spaces

“…In this subsection, we introduce those notions and prove the necessary results so we can obtain Theorem 1.4. Similar properties have already appeared in the literature, and we refer the interested reader to [11] and [12] for a detailed discussion about transfinite asymptotic properties and their relation with the Szlenk index.…”

Higher projective tensor products of $c_0$

“…However, the boundedness of the quotient map does not provide any analogue of the inequality s t y s 1 in the pulled back collection. 6. Extension to the symmetric case Lemma 6.1.…”

“…In [6], an analogous stratification is discussed. For a given natural number n and exponent p ∈ (1, ∞], a property A n,p is described.…”