On Closed Subspaces of Non-Archimedean Nuclear Fréchet Spaces with a Schauder Basis

Abstract

A complete valued field (K,|⋅|) is non-Archimedean if its valuation satisfies the strong triangle inequality: |a+b|≤max{|a|,|b|} for all a,b∈K. In this paper all Frechet spaces are over a non-Archimedean field \(\mathbb{K}\). We prove that every nuclear Frechet space is isomorphic to a closed subspace of some nuclear Frechet space with a Schauder basis. Next we show that the class of all nuclear Frechet spaces has no universal element (i.e., there exists no nuclear Frechet space E such that every nuclear Frechet space is isomorphic to a closed subspace of E). Finally we prove that a nuclear Frechet space F is isomorphic to a closed subspace of some nuclear Kothe space (i.e., some Frechet space with a Schauder basis and with a continuous norm) if and only if F is countably normed.