Locally Convex Cones

The purpose of this chapter is twofold: Firstly, to provide the tools and the settings for the integration theory which will be developed in Chapters II and III, and secondly, to introduce the theory of locally convex cones to a wider audience. This theory generalizes locally convex topological vector spaces and has (in the author’s opinion, quite unsurprisingly) not yet received the attention that it deserves. Locally convex cones permit many more and substantially different examples and applications than locally convex vector spaces. In the aspects of the theory that have been developed so far, the increase in generality leads only to minor, if any at all, compromises with respect to the depth of its results. While some of the methods and arguments employed may at times appear rather technical and indeed counterintuitive, this is largely the consequence of the inclusion of infinity-type unbounded elements and the general non-availability of the cancellation law.

So why is it worth the effort? Endowed with suitable topologies, vector spaces yield rich and well-studied structures. Locally convex topological vector spaces permit an extensive duality theory whose study gives valuable insight into the spaces themselves. Some important mathematical settings, however, while close to the structure of vector spaces do not allow subtraction of their elements or multiplication by negative scalars. Examples are certain classes of functions that may take infinite values or are characterized through inequalities rather than equalities. They arise naturally in integration theory, potential theory and in a variety of other settings. Likewise, families of convex subsets of vector spaces which are of interest in various contexts, do not form vector spaces. If the cancellation law fails, domains of this type can not be embedded into vector spaces in order to apply the results and techniques from classical functional analysis. The inclusion of these and similar examples into an analytical theory merits the investigation of a more general structure. Apart from being useful in this sense, the theory of locally convex cones allows for some interesting and occasionally insightful and elegant mathematics.