Topological structures in Colombeau algebras: topological ℂ̃-modules and duality theory

We study modules over the ring ℂ̃ of complex generalized numbers from a topological point of view, introducing the notions of ℂ̃-linear topology and locally convex ℂ̃-linear topology. In this context particular attention is given to completeness, continuity of ℂ̃-linear maps and elements of duality theory for topological ℂ̃-modules. As main examples we consider various Colombeau algebras of generalized functions. © Springer 2005.