A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces (Q2773381)

Classifying Banach spaces, or defining in simple terms important families of Banach spaces, is difficult. This article explains why, by displaying a proper frame which allows evaluating the topological complexity of the classical isomorphism classes of Banach spaces. Such results were somehow implicit in previous works, such as J. Bourgain's remarkable proof of the fact that the family of Banach spaces with separable dual has no universal element. This article represents, however, an important progress, since it allows the direct use of topological tools (such as the Kunen-Martin uniform boundedness principle) for obtaining positive results.NEWLINENEWLINENEWLINESection 1 of this article contains the construction, which follows the lines of James and Lindenstrauss-Stegall's constructions, of Banach spaces built on trees. Effros-Borel structure is used in Section 2 for building the frame in which one can properly speak of Borel structure on the ``set'' of separable Banach spaces. It is shown, in particular, that there are no Borel calculable invariants which classify separable Banach spaces up to isomorphism.NEWLINENEWLINENEWLINEMany natural classes of Banach spaces, e.g., the class of reflexive spaces, are shown to be coanalytic non-Borel in Section 3, and natural ranks (ranks of embedding, Szlenk index,\dots) are shown to be coanalytic ranks (or norms) on these families in Section 4. Section 5 undertakes a similar work on the coding of basic sequences and of the equivalence between bases. The corresponding equivalence relation has no analytic section, and the families of shrinking or boundedly complete bases are coanalytic non-Borel. Finally, Section 6 uses T. Gowers's negative solution to the hyperplane problem for providing an embedding of the equivalence relation denoted \(E_0\) into the isomorphism relation between Banach spaces. It follows from the existence of such an embedding that this relation has no analytic section.