On the absolute convergence of series in topological spaces

We consider a vector space over a field of real numbers; under a topological vector space we understand a Hausdorff topological vector space. The topology in this space is defined by the basis of the neighborhoods of zero that satisties the Fon Niemann axioms. Here we study the criteria for absolutely ρ-convergence of numerical series in a topological vector space. It is proved thatthe set of all absolutely ρ-convergent (0