Descriptive complexity of some isomorphism classes of Banach spaces

We present a topological frame in which it is possible to estimate the complexity of some Borel families of separable Banach spaces. We use this frame for evaluating the complexity of the isomorphism class of the Hilbert space l(2), of certain asymptotically Hilbertian spaces, and of the l(p) spaces under the condition 1 < p < infinity. The complexity appears to increase with the distance to the Hilbert space. Commented open problems conclude the article. (C) 2018 Published by Elsevier Inc.