Tsirelson-like spaces and complexity of classes of Banach spaces

TL;DR

This paper demonstrates the complexity of classifying Banach spaces by showing that certain classes, like subspaces of c0, have high descriptive set-theoretic complexity, using Tsirelson-like space constructions.

Contribution

It establishes the descriptive set-theoretic complexity of classes of Banach spaces, including subspaces of c0, Schur property, and Dunford-Pettis property, using Tsirelson-like space constructions.

Findings

Class of subspaces of c0 is a complete analytic set.

Classes with Schur and Dunford-Pettis properties are -complete.

Uses Tsirelson-like space constructions to analyze complexity.

Abstract

Employing a construction of Tsirelson-like spaces due to Argyros and Deliyanni, we show that the class of all Banach spaces which are isomorphic to a subspace of $c_{0}$ is a complete analytic set with respect to the Effros Borel structure of separable Banach spaces. Moreover, the classes of all separable spaces with the Schur property and of all separable spaces with the Dunford-Pettis property are $\mathbf{\Pi}^{1}_{2}$-complete.