Nonclosed sequentially closed subsets of locally convex spaces and applications

Abstract

This article is a review of some methods of constructing nonclosed sequentially closed subsets in locally convex spaces (l.c.s.) as well as some applications of such subsets to problems in the theory of l.c.s.These subsets are collections of elements having two or more indexes being natural numbers, as well as convex or linear envelopes of such countable sets. As to the above-mentioned problems we regard several ones connected with the Ptak and Krein-Smulian spaces (we recall the definitions of these spaces below), problems connected with the theory of differentiable functions on l.c.s. and some problems posed by Dieudonné and L. Schwartz and solved by Grothendieck (in the latter case we give solutions which differ from the solutions of Grothendieck).In some cases we prefer not to give the most general constructions replacing them by typical examples.