Topological structures in Colombeau algebras: topological $\tilde{\text{C}}$-modules and duality theory

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