Asymptotic smoothness and universality in Banach spaces
Volume 172 / 2023
Colloquium Mathematicum 172 (2023), 281-324
MSC: Primary 46B20; Secondary 46B03, 46B06.
DOI: 10.4064/cm8923-12-2022
Published online: 28 February 2023
Abstract
For $1 \lt p\leq \infty $, we study the complexity and the existence of universal spaces for two classes of separable Banach spaces, denoted $\mathsf{A}_p$ and $\mathsf{N}_p$, and related to asymptotic smoothness in Banach spaces. We show that each of these classes is Borel in the class of separable Banach spaces. Then we build small families of Banach spaces that are both injectively and surjectively universal for these classes. Finally, we prove the optimality of this universality result, by showing in particular that none of these classes admits a universal space.
