Compatible uniformities and closed mappings

Abstract

Gel'fand and Šilov(1) have introduced the concept of compatibility of norms on a vector space, defining two norms to be compatible if each sequence which is Cauchy with respect to both norms and converges to zero with respect to one norm, also converges to zero with respect to the other. They have found it convenient to build up their exposition of spaces of generalized functions (2) by developing the theory of locally convex spaces whose topology can be defined by a countable set of pairwise compatible norms. The author (3) has examined the properties of locally convex spaces which have been topologized by making a set of linear mappings into locally convex spaces continuous, and has shown that if this method is applied to a set of closed operators on Hilbert space, then the resulting spaces have topologies generated by a set of pairwise compatible norms. The purpose of this paper is to generalize this result by putting it in its natural context as a theorem on uniformities which have been defined by making a closed set of mappings uniformly continuous. The closed mappings of the title are mappings with closed graph.