Polish spaces of Banach spaces

TL;DR

This paper studies natural Polish spaces of separable Banach spaces defined via norms on rational vector spaces, comparing their topological and Borel complexity properties and analyzing generic features.

Contribution

It introduces and thoroughly analyzes Polish spaces of Banach spaces, comparing them with existing admissible topologies and exploring their Borel complexities and generic properties.

Findings

Borel complexities differ little between the two topological approaches.

Generic properties depend on the choice of admissible topology.

The spaces of norms on rational vector spaces are natural and well-structured.

Abstract

We present and thoroughly study natural Polish spaces of separable Banach spaces. These spaces are defined as spaces of norms, resp. pseudonorms, on the countable infinite-dimensional rational vector space. We provide an exhaustive comparison of these spaces with admissible topologies recently introduced by Godefroy and Saint-Raymond and show that Borel complexities differ little with respect to these two different topological approaches. We investigate generic properties in these spaces and compare them with those in admissible topologies, confirming the suspicion of Godefroy and Saint-Raymond that they depend on the choice of the admissible topology.