The identity $L(E,F) = LB(E,F)$, tensor products and inductive limits

Abstract

The purpose of this article is to extend the study of pairs of locally convex spaces (E, F) such that every continuous linear map from E into F is bounded (i.e. maps a 0-nghb in E in a bounded subset of F), and to give applications of our results in this direction to the interchangeability of induttive limits and projective or injective tensor products and to the projec-tive description of weighted induttive limits of spaces of vector valued continuous functions. Our investigations are related to the study of certain classes of Fréchet spaces which have been relevant in several areas recently (see [2], [3], [13], [24], [25], [37], [45]).