Duality Theory

Abstract

Assuming that the nonarchimedean field K is spherically complete we develop in this chapter two concepts which are at the base of any deeper investigation into locally convex vector spaces. As explained in§12 the role of compact and precompact subsets is taken over by c-compact and compactoid o-sub-modules, respectively, in a locally convex K-vector space. Roughly speaking these are o-linear versions of the former topological properties. The concept of polarity which we describe in§13 is the fundamental tool for duality theory. The formation of the pseudo-polar passes from o-submodules in a given locally convex K-vector space V to o-submodules in the continuous linear dual V′ thereby allowing for the transfer of information back and forth. It is a crucial technical detail in the definition of the pseudo-polar to only consider those continuous linear forms whose absolute value on the given o-submodule is strictly less than one. Relaxing this condition to less than or equal to one would lead to the more traditional notion of the “polar”. In nonarchimedean functional analysis this latter notion does not work so well and therefore is not treated in this book at all. In§14 we introduce and study the class of admissible topologies on a given locally convex vector space. These consist of those locally convex topologies which give rise to the same continuous linear dual as the given topology. There is a weakest and a finest such topology — the weak and the Mackey topology.