Boundedness in linear topological spaces

Abstract

form a base at 0. In ?1 we give a necessary and sufficient condition, in terms of invariant pseudometrics, for a linear topological space to be locally bounded. In ?2 we discuss the relationship of our results with other results known on the subject. In ?3 we introduce two ways of classifying the locally bounded spaces into types in such a way that each type contains exactly one of the IP spaces (0 < p < 1), and show that these two methods of classification turn out to be identical. Also in ?3 we prove a metrization theorem for locally bounded spaces, which is related to the normal metrization theorem for uniform spaces, but which uses a different induction procedure. In ?4 we introduce a large class of linear topological spaces which includes the locally convex spaces and the locally bounded spaces, and for which one of the more important results on boundedness in locally convex spaces is valid.