Abstract
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 ℓ2, of certain asymptotically Hilbertian spaces, and of the ℓp spaces under the condition 1<p<∞. The complexity appears to increase with the distance to the Hilbert space. Commented open problems conclude the article.
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