We study topological vector spaces over the field $P$ of real or complex numbers, endowed with the discrete topology. These objects are called topological vector groups (for brevity, TVGs). By the conjugate$E'$of a locally convex TVG$E$we mean the set of all continuous linear mappings of$E$into$P$, where$P$is equipped with the usual (for the plane or the line) topology. We construct a duality theory for locally convex TVGs. In particular, we obtain an analog of the Mackey–Arens Theorem: in$E$there exists the strongest locally convex TVG topology compatible with the duality between$E$and$E'$. This topology is the topology of uniform convergence on all absolutely convex, weakly complete subsets of$E'$. Each such subset is the product of a weakly compact, absolutely convex set by a weakly complete subspace (that is, by a product of lines).In the present article we also study the connection between weakly complete subsets of a TVG and the subsets satisfying “the double limit condition”. The results are applied to give a proof of Eberlein's Theorem for locally convex TVGs. In addition, we prove that a subset satisfying “the double limit condition” in the strict inductive limit of complete, locally TVGs is necessarily contained in some limiting space.Bibliography: 8 titles.