Three-space-problems and small ball properties for Fréchet spaces
\textit{L. Frerick} and \textit{A. Peris} [RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 97, No.~2, 257--261 (2003; Zbl 1068.46001)] introduced the following concept: let \(\mathcal A\) be a family of subsets of a metric space \(X\); a subset \(D \subset X\) has the \(\mathcal A\)-small ball property if for every sequence \(\varepsilon_n > 0\) tending to 0 monotonically, there is a sequence of \(A_n \in \mathcal A\) such that \(D \subset \bigcup_{n \in \mathbb N}\{x \in X: d(x, A_n) < \varepsilon_n\}\). Several properties of Fréchet spaces are characterized in terms of \(\mathcal A\)-small ball properties. In the paper under review, the author presents a unified approach to three-space properties in the context of Fréchet spaces by using \(\mathcal A\)-small ball properties. In particular, this gives a new proof of the fact that the properties of being a Schwartz space, a quasinormable space, a Montel space and reflexivity are three-space properties. The paper is dedicated to the memory of Susanne Dierolf.
scientific article; zbMATH DE number 5914712