Topological totally convex spaces I

Abstract

Following suggestions of Pumplün and Röhrl we study topologies on totally convex spaces that are generated by families of semi-norms. In particular, we generalize Alaoglu's Theorem by showing that, for any totally convex spaceX, the initial topology on the dual space hom (X, Ô C) for the evaluationse(x) in the pointsx ofX is a compact topology. With the help of topological methods we also obtain a transparent description of the left adjointS of the unitball functorô:Ban ₁ →TC. The functorS has been identified before by Pumplün and Röhrl in an algebraic context using many nontrivial arguments and computations.