Descriptive set theory of separable Fréchet spaces
In the past few decades, much has been done regarding the descriptive set theory of separable Banach spaces. However, the descriptive properties of separable Fréchet 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 Fréchet spaces such as Fréchet-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 Fréchet 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.