Geodesic
On topological vector groups
Sbornik. Mathematics, Tome 10 (1970) no. 4, pp. 531-546 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {P. S. Kenderov},
     title = {On~topological vector groups},
     journal = {Sbornik. Mathematics},
     pages = {531--546},
     year = {1970},
     volume = {10},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_10_4_a3/}
}
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P. S. Kenderov. On topological vector groups. Sbornik. Mathematics, Tome 10 (1970) no. 4, pp. 531-546. http://geodesic.mathdoc.fr/item/SM_1970_10_4_a3/

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