Polish spaces of Banach spaces

Abstract

We present and thoroughly study natural Polish spaces of separable Banach spaces. These spaces are defined as spaces of norms, respectively 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 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.

Polish spaces of Banach spaces. Complexity of isometry and isomorphism classes

“…In [14] we compared our spaces P ∞ and B with (SB, τ ). In particular, we observed that whenever τ is an admissible topology, then there exists a continuous map Φ : (SB, τ ) → P such that for every F ∈ SB(X) we have F ≡ X Φ(F ) isometrically, see [14,Theorem 3.3]. Thus, from our results obtained in the coding P ∞ one may easily deduce also results formulated in the language of admissible typologies.…”

“…Finally, in order to avoid any confusion, we emphasize that if we write that a mapping is an "isometry" or an "isomorphism", we do not mean it is surjective if this is not explicitly mentioned. [14]. The most important notion we want to recall is the Polish spaces of Banach spaces.…”

“…This drawback was addressed in a recent paper [28] of Godefroy and Saint-Raymond where they propose to study certain natural topologies on the standard Borel spaces of separable Banach spaces and they compute Borel complexities, in these topologies, of several important classes of Banach spaces. The authors of this paper initiated in [14] the study of the Polish spaces of norms and pseudonorms on the countable infinite-dimensional vector space over Q, which have additional further advantages:…”

“…• the computation of Borel complexities of various classes of separable Banach spaces is usually in these spaces as straightforward as possible, which in particular allows us to improve several estimates by Godefroy and Saint-Raymond; • this approach to topologizing the space of Banach spaces is somewhat similar to how metric structures are topologized generally in continuous model theory, which could connect descriptive set theory of Banach spaces with the model theory of Banach spaces, which is one of the most developed areas of applications of logic to metric structures. This paper is a companion and second part to [14], however it is completely selfcontained and can be read independently. Its aim is to demonstrate the strength of this new approach by computing (in many cases, these computations are sharp) complexities of several classes of Banach spaces, focusing on isomorphism classes, improving several results of Godefroy and Saint-Raymond, and initiating the research of computing the complexities of isometry classes.…”

“…

We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces.We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is Fσ.

…”