On the complexity of some inevitable classes of separable Banach spaces

In this paper, we study the descriptive complexity of some inevitable classes of Banach spaces. Precisely, as shown in [11], every Banach space either contains a hereditarily indecomposable subspace or an unconditional basis, and, as shown in [8], every Banach space either contains a minimal subspace or a continuously tight subspace. In this note, we study the complexity of those inevitable classes as well as the complexity of containing a subspace in any of those classes.