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).
