On the Banach-Steinhaus Theorem and the Continuity of Bilinear Mappings

Abstract

In this paper, the BANACH-STEINHAUS theorem is extended from its usual locally convex topological vector space setting to the much broader framework of convergence vector spaces. It is used to derive theorems yielding the joint continuity of separately continuous bilinear mappings. These results are used, in turn, to show that the convolution mapping is a jointly continuous bilinear mapping when the distribution spaces ℰ and carry the canonical convergence vector space structures.