International Journal of Mathematics and Mathematical Sciences
Volume 12 (1989), Issue 4, Pages 625-632

The closed neighborhood and filter conditions in solid sequence spaces

P. D. Johnson, Jr.

Department of Algebra, Combinatorics, and Analysis, 120 Math Annex, Auburn University, Auburn 36849, Alabama, USA

Received 30 June 1988

Copyright © 1989 P. D. Johnson, Jr.. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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).