Some locally convex spaces of continuous vector-valued functions over a completely regular space and their duals

Abstract

The strict, superstrict and the β F {\beta _F} topologies are defined on a space A of continuous functions from a completely regular space into a Banach space E. Properties of these topologies are discussed and the corresponding dual spaces are identified with certain spaces of operator-valued measures. In case E is a Banach lattice, A becomes a lattice under the pointwise ordering and the strict and superstrict duals of A coincide with the spaces of all τ \tau -additive and all σ \sigma -additive functionals on A respectively.