Duality of topological modules over normed rings

In this paper, we introduce the operator topology for the set of all continuous homomorphisms between two topological modules, and discuss the duality of topological modules over an admissible normed ring R (see Definition 2.1). We show that the dual functor B(−,R) defined on the category of locally bounded R-modules is topologically left exact. Moreover, if R is complete, and the modules are extendible, the dual functor is topologically exact.