The closed neighborhood and filter conditions in solid sequence spaces

Abstract

Let E be a topological vector space of scalar sequences, with topology τ; (E,τ) satisfies the closed neighborhood condition iff there is a basis of neighborhoods at the origin, for τ, consisting of sets whlch are closed with respect to the topology π of coordinate‐wise convergence on E; (E,τ) satisfies the filter condition iff every filter, Cauchy with respect to τ, convergent with respect to π, converges with respect to τ.Examples are given of solid (definition below) normed spaces of sequences which (a) fail to satisfy the filter condition, or (b) satisfy the filter condition, but not the closed neighborhood condition. (Robertson and others have given examples fulfilling (a), and examples fulfilling (b), but these examples were not solid, normed sequence spaces.) However, it is shown that among separated, separable solid pairs (E,τ), the filter and closed neighborhood conditions are equivalent, and equivalent to the usual coordinate sequences constituting an unconditional Schauder basis for (E,τ). Consequently, the usual coordinate sequences do constitute an unconditional Schauder basis in every complete, separable, separated, solid pair (E,τ).