On topological vector groups

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.