Spaces of Linear and Bilinear Mappings

Abstract

The set L (E, F) of all continuous linear mappings of E in F, where both E and F are locally convex, is a vector space. If F = K, then L (E, F) = E′ and so it is obvious that there are many possibilities to define a locally convex topology on L (E, F). This is done in § 39 and by adapting the methods of Volume I it is possible to obtain generalizations of some classical theorems as the Banach-Mackey theorem and the Banach-Steinhaus theorem. The relation between equicontinuous and weakly compact subsets of L (E, F) is a little more complicated than in the case of dual spaces.