Counter Examples in Non-Archimedean Locally Convex Spaces
M. E. Egwe, J. A Braimah
TL;DR
This paper investigates the existence and structure of Schauder bases and related decompositions in non-archimedean locally convex and Fréchet spaces, revealing counterexamples that challenge naive generalizations from classical spaces. It develops and leverages concepts such as $t$-orthogonality, Schauder partitions, and strong finite-dimensional decompositions to analyze when subspaces possess or lack Schauder bases, including normability considerations. The authors show that, unlike finite-type Fréchet spaces which behave like $K^N$ with all closed subspaces having bases, many infinite-dimensional Fréchet spaces of countable type do not admit a Schauder basis, and even spaces with decompositions can lack a basis. These results illuminate the intricate subspace landscape in non-archimedean settings and clarify the limits of basis-based decompositions in such spaces.
Abstract
In this paper, we shall consider some counter examples in non-archimedean locally convex spaces with special closed subspaces and Schauder basis in non-archimedean Fréchet spaces as well as closed subspaces \emph{without} Schauder basis in non-archimedean Fréchet spaces.