On Convex Topological Linear Spaces

Abstract

Introduction. In an earlier article [911 the author developed at some length the theory of certain mathematical objects which he called linear systems. It is the purpose of the present paper to apply this theory to the study of convex topological linear spaces. This application is based on the many-to-one correspondence between convex topological linear spaces and linear systems which may be set up by assigning to each such space Xt the linear system XL where X is the abstract linear space underlying Xt and L is the set of all continuous linear functionals on X. First of all the nature of the family of all Xe's belonging to a given XL is studied and it is shown that it has a weakest and a strongest member. The bulk of the paper is then devoted to correlating the properties of a convex topological linear space with those of its linear system and with the strength of its topology relative to that of the others with the same linear system. A general survey of the contents of this paper will be found in [10]. 1. Preliminary definitions and remarks. In this section we recall briefly some of the notions from the theory of topological linear spaces and from [9] which we shall need in the sequel. By a topological linear space(2) we mean a real linear space which is at the same time a T1 space in the sense of Alexandroff and Hopf [1I and in which the topology is related to the algebra in such a manner that the operations of addition and multiplication by reals are continuous in both variables together. By a convex topological linear space(2) we mean a topological linear space in which every point has a complete system of convex neighborhoods. By a linear functional on a linear space we mean a function I defined from the space to the reals such that 1(Xx+,yy) =X(x) +,.t(y) for all x and y in the space and all real numbers X and u. If X is a linear space we denote by X* the linear space of all linear functionals on X. By a linear system XL we mean a linear space X together with a distinguished