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Examples of non-Archimedean twisted nuclear Fréchet spaces. (English) Zbl 1134.46051

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.

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46A04 Locally convex Fréchet spaces and (DF)-spaces
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)

Citations:

Zbl 1016.46047
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Full Text: Euclid