Descriptive set theory of separable Fr\'echet spaces

TL;DR

This paper investigates the descriptive set theory of separable Fréchet spaces, analyzing their complexity and classifying specific subclasses like Montel spaces, revealing new insights into their structural properties.

Contribution

It extends the descriptive set theory framework from Banach spaces to Fréchet spaces, computing the complexity of classical classes and solving an old problem about Montel spaces.

Findings

Montel spaces form a complete coanalytic class.

No separable Montel space contains isomorphic copies of all others.

Descriptive complexity estimates for various classical Fréchet spaces.

Abstract

In the past few decades, much has been done regarding the descriptive set theory of separable Banach spaces. However, the descriptive properties of separable Fr\'echet spaces have not yet been investigated. In these notes, we look at this problem, its relation with the (now standard) theory for separable Banach spaces, and we compute/estimate the descriptive complexity of some classical classes of separable Fr\'echet spaces such as Fr\'echet-Hilbert, Schwartz, nuclear, and Montel spaces. Our main result shows that the class of Montel spaces is complete coanalytic. Noticeably, this applies outside the realm of descriptive set theory and solves an old problem regarding Fr\'echet spaces satisfying the Heine--Borel property (i.e., Montel spaces). Precisely, we show that there is no separable Montel space containing isomorphic copies of all separable Montel spaces.