On the symmetry and bounded closure of locally convex spaces

Abstract

Preface. For some time it has been recognized that the properties of symmetry and bounded closure are of cardinal importance in the theory of locally convex spaces. The purpose of this paper is to discuss symmetry and bounded closure with the aid of a new topology, and to discuss the new topology itself. The introduction includes the preliminaries to the rest of the paper. The account of the basic theory of locally convex spaces is fairly complete, but brief. We have given proofs only when they represent some contribution. As the references indicate, most of the theorems of the introduction can be found in the literature. In ?1 we introduce the new topology, called the GF topology, and develop the theory of the GF space. A space is a GF space if and only if it is symmetric and boundedly closed. Study of the GF topology on duals leads to information about interior maps and quotient spaces, information which is not obvious from direct consideration of symmetry and bounded closure, We conclude the section with several examples. In view of the general results of the first section it is important to know what spaces are GF spaces. In ?2 we take up this question for direct products and their duals, and also the questions of symmetry and bounded closure. ?3 serves the same purpose for continuous function spaces. Finally in ?4 we discuss metric and LF spaces and their duals. The influence of the works of G. W. Mackey, J. Dieudonne, and L. Schwartz is apparent throughout the paper, and it is to these authors that we owe our interest in locally convex spaces. The terminology used in the paper is almost exclusively that employed in the works of Dieudonne, Schwartz, and Bourbaki. For the convenience of the reader we list the exceptions. We distinguish between weak and weak-star topologies; our total bounded is the French precompact and corresponds to the definition given in [14]; we owe the term relatively strong to Mackey [13]; where the French authors use filters we make use of nets, as in Kelley [10]; and for a uniform structure topology the natural definition of a Cauchy net leads to a concept of completeness which is equivalent to the French term and stronger than that given in [14]. There are no French equivalents for symmetry and bounded closure.