A New Class of Locally Convex Spaces with application

In this paper we define the Radon-Nikodym class (RN class) of locally convex topological vector spaces. The RN class is characterized in terms of the Radon-Nikodym theorem for vector measures using integrable by seminorm derivatives. It is shown that the RNP class of all complete Hausdorff locally convex spaces possessing the Radon-Nikodym property is properly contained in the RN class. As an application we present a Radon-Nikodym theorem for multimeasures with respect to the Pettis integrable multifunctions in terms of the RN class.