Three and a half asymptotic properties
Volume 257 / 2021
Studia Mathematica 257 (2021), 155-212
MSC: Primary 46B03; Secondary 46B06.
DOI: 10.4064/sm190528-23-5
Published online: 10 September 2020
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 quasi-reflexive, asymptotically $p$-smoothable Banach spaces to quasi-reflexive Banach spaces with asymptotic equal norm type $p$.
