On closed subspaces of non-Archimedean nuclear Fréchet spaces with a Schauder basis (Q2256827)

The author continues the study that he started in 2000 about the structure of Fréchet spaces \(E\) over non-Archimedean valued fields. This time he pays attention to closed subspaces of non-Archimedean nuclear Fréchet spaces with a Schauder basis. Let \(\Gamma\) be the family of all non-decreasing unbounded sequences of real positive numbers. First he proves that every nuclear Fréchet space \(E\) is isomorphic to a closed subspace of \([A_{\infty}(\beta)]^{\mathbb{N}}\) for some power series space of infinite type \(A_{\infty}(\beta)\), \(\beta \in \Gamma\). He also gives a (typically Archimedean) proof of the classical counterpart of this result, which was hither unknown . Next he shows that for every nuclear Fréchet space \(E\) there exists \(\alpha \in \Gamma\) such that \(E\) has no closed subspace isomorphic to \(A_{\infty}(\alpha)\). Hence, the class of all non-Archimedean nuclear Fréchet spaces has no universal element. Finally, he proves that a nuclear Fréchet space \(E\) is isomorphic to a closed subspace of some nuclear Köthe space if and only if \(E\) is countably normed (recall that a Köthe space is a Fréchet space with a Schauder basis and with a continuous norm). A similar result was proved by \textit{L. Holmström} [Proc. Am. Math. Soc. 89, 453--456 (1983; Zbl 0532.46001)] and by \textit{D. Vogt} and \textit{V. Walldorf} [Arch. Math. 61, No. 5, 459--464 (1993; Zbl 0814.46003)] for Fréchet spaces over \(\mathbb{R}\) or \(\mathbb{C}\).