On the complexity of some inevitable classes of separable Banach spaces

TL;DR

This paper investigates the descriptive complexity of certain unavoidable classes of Banach spaces, analyzing the complexity of their subspace structures and the conditions under which they contain specific types of subspaces.

Contribution

It provides a detailed complexity analysis of classes of Banach spaces characterized by containing particular subspaces, extending previous results on their structural properties.

Findings

Identifies the descriptive complexity of classes containing hereditarily indecomposable or minimal subspaces.

Analyzes the complexity of classes with unconditional bases or tight subspaces.

Provides insights into the subspace structure of Banach spaces based on complexity measures.

Abstract

In this paper, we study the descriptive complexity of some inevitable classes of Banach spaces. Precisely, as shown in [Go], every Banach space either contains a hereditarily indecomposable subspace or an unconditional basis, and, as shown in [FR], every Banach space either contains a minimal subspace or a continuously tight subspace. In these notes, we study the complexity of those inevitable classes as well as the complexity of containing a subspace in any of those classes.