| Résume | We introduce a certain Polish space of all separable Banach spaces. The definition is in the same spirit as Grigorchuk's space of marked groups or (slightly less well known) Vershik's space of all separable complete metric spaces. We compare it with a recent different approach to topologizing the space of separable Banach spaces, by Godefroy and Saint-Raymond. Our main interest will be in the descriptive complexity of classical Banach spaces with respect to this Polish topology. We show that the separable infinite-dimensional Hilbert space is characterized as the unique Banach space whose isometry class is closed, and also as the unique Banach space whose isomorphism class is F_sigma, where the former employs the Dvoretzky theorem and the latter the solution to the homogeneous subspace problem. For p in [1,infty)-{2}, we mention that the isometry class of L^p[0,1] is G_delta-complete and the class of l^p is F_sigma,delta-complete. Also, the isometry class of c_0 is F_sigma,delta-complete. The talk will be aimed at an audience with basic knowledge of descriptive set theory and general topology, but no particular knowledge of Banach space theory. It will be based on joint work with Marek Cuth, Martin Dolezal and Ondrej Kurka. |
