When the set of closed subspaces of C(Δ), where Δ is the Cantor set, is equipped with the standard Effros-Borel structure, the graph of the basic relations between Banach spaces (isomorphism, being isomorphic to a subspace, quotient, direct sum,...) is analytic non-Borel. Many natural families of Banach spaces (such as reflexive spaces, spaces not containing ℓ₁(ω),...) are coanalytic non-Borel. Some natural ranks (rank of embedding, Szlenk indices) are shown to be coanalytic ranks. Applications are given to universality questions. Analogous results are shown for basic sequences modulo equivalence.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm172-2-3,
author = {Benoit Bossard},
title = {A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {117-152},
zbl = {1029.46009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm172-2-3}
}
Benoit Bossard. A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces. Fundamenta Mathematicae, Tome 173 (2002) pp. 117-152. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm172-2-3/