An Isometric Representation of the Dual of $ {\user1{\mathcal{C}}}{\left( {X,\mathbb{R}} \right)} $

Abstract

Many structures in functional analysis are introduced as the limit of an inverse (aka projective) system of seminormed spaces [2,4,9]. In these situations, the dual is moreover equipped with a seminorm. Although the topology of the inverse limit is seldom metrizable, there is always a natural overlying locally convex approach structure. We provide a method for computing the adjoint of this space, by showing that the dual of a limit of locally convex approach spaces becomes a co-limit in the category of seminormed spaces. As an application we obtain an isometric representation of the dual space of real valued continuous functions on a locally compact Hausdorff space X, equipped with the compact open structure. (2000): 46M40, 46A03, 46E27, 28A33. Mathematics Subject Classifications