Examples of non-Archimedean twisted nuclear Fréchet spaces (Q2469112)

A Fréchet space is called twisted if it is not isomorphic to a countable product of Fréchet spaces with continuous norms. It is shown that no non-Archimedean Fréchet space with a Schauder basis is twisted (Proposition 1). By constructing examples of non-Archimedean twisted nuclear Fréchet spaces, it is proved that the answer to the problem whether any Fréchet space of countable type is isomorphic to a countable product of Fréchet spaces with continuous norms, is negative even for nuclear Fréchet spaces (Theorem 7 and Proposition 8). The author constructed in [Indag. Math., New Ser. 11, No. 4, 607--616 (2000; Zbl 1016.46047)] many examples of Fréchet spaces of countable type without a Schauder basis, but each of these spaces is a countable product of Fréchet spaces with continuous norms.