Logic Seminar

The talk will consist of two parts. In the first one, we introduce a certain natural Polish space of all separable Banach spaces. We compare it with the 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 Fσ, where the former employs the Dvoretzky theorem and the latter the solution to the homogeneous subspace problem. For p in [1,∞)−{2}, we mention that the isometry class of Lp[0,1] is Gδ-complete and the class of ℓp is Fσ,δ-complete.
In the second part, we connect it with the recent study of Fraïssé Banach spaces, initiated by Ferenczi, López-Abad, Mbombo, and Todorčević. We show that a Banach space has a comeager isometry class in its closure if and only if it is the unique limit of a 'weak Fraïssé class' of finite-dimensional spaces. While it is open whether there are other Fraïssé Banach spaces besides the Gurariĭ space and the Lp's, we show there are more examples of generic Banach spaces in the weaker sense above.
The first part will be based on joint work with M. Doležal, M. Cúth, and O. Kurka; the second is a work in progress jointly with M. Cúth and N. de Rancourt.