Group topologies on vector spaces and character lifting properties

It is known that every continuous character on a topological vector space can be lifted to a continuous linear functional and, moreover, these liftings give rise to a topological isomorphism between the dual group and the dual space, when both are endowed with the compact-open topology. We investigate the presence of these properties in more general topologized real vector spaces. 1. Preliminaries generalizations of several basic theorems from abstract harmonic analysis and duality theory of topological Abelian groups, which were carried out by himself and other authors. The theory makes essential use of locally convex vector groups and other classes of topologized vector spaces. In this paper we rst give a survey of different topological properties that may be present in a topological Abelian group which algebraically is a real vector space. Next we explore the way these properties are related to the possibility of lifting continuous characters to continuous linear functionals on these spaces. Note that in the topological group framework, Nickolas ((18)) showed that the related problem of lifting continuous characters to continuous real valued group homomorphims is relevant in the study of the structure of dual groups.