The subject of the present work deals with integral representations of bounded operators, acting on linear spaces of vector valued functions.

First we will consider operators on the space  of all continuous functions  vanishing at infinity, endowed with the uniform topology, S being a locally compact space and X a Banach space. We will give a complete characterization of operators  which enjoy an integral form with respect to a scalar measure m on S.

Next we consider integral representations for operators on  type spaces, with values in a Banach space or a locally convex space. The main setting is the Bochner integration process with respect to finite abstract measure. The integral representations obtained may be considered as generalizations of the classical Riesz Theorem.

integral representation, bounded operators, Bochner integration process, vector measures.