Orthogonal and Schauder bases in non-archimedean locally convex spaces (Q2751728)

Let \((E,\mathcal P)\) be a non-archimedean (n.a.) locally convex space over a complete n.a. valued field \(\mathbb K\), where \(\mathcal P\) is a family of n.a. seminorms on \(E\) generating the topology [see \textit{A. C. M. van Rooij}, ``Non-Archimedean Functional Analysis'', New York (1978; Zbl 0396.46061)]. A sequence \((x_n)\) in \(E\) is called orthogonal if \(p(\sum_{i=1}^n\lambda_ix_i) = \max_{1\leq i\leq n}p(\lambda_ix_i),\) for all \(\lambda _i \in \mathbb K\), all \(n\in \mathbb N\), and all \(p\in \mathcal P\). The sequence \((x_n)\) is called a topological base of \(E\) if every \(x\in E\) can be written as \(x=\sum_{n=1}^\infty \lambda_nx_n\), and a Schauder base if, further, the coefficient functionals \(f_n(x) =\lambda _n\) are continuous. The paper is concerned with various results relating properties of \(E\) (Orlicz-Pettis property, Montel property, metrizability, etc.) with the existence of orthogonal and basic sequences. For instance, every orthogonal sequence is basic, i.e. a Schauder base for its closed linear hull. An open problem is whether any Fréchet space of countable type has a Schauder base. The authors give a partial answer to this question showing that this is true in the case when all quotient spaces \(E/\)ker\(p, p\in \mathcal P\), are finite dimensional. Recently, \textit{W. Sliwa} [to appear in Bull. Beg. Math. Soc. -- Simon Stevin 8, No. 1, 109-118 (2001)] proved that every Fréchet space of countable type contains a basic orthogonal sequence. Section 5 is concerned with compactoids, meaning subsets \(A\) of \( E \) such that for every 0-neighborhood \(U\) in \(E\) there is a finite set \(Z\subset E\) with \(A\subset U +\operatorname {co}(Z)\). In this section some properties valid in the case of Banach spaces are extended to locally convex spaces. The paper contain many other interesting results.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00058].