Foundations

Abstract

In this chapter we introduce the basic notions and constructions of nonar-chimedean functional analysis. We begin in §1 with a brief but self-contained review of nonarchimedean fields. The main objective of functional analysis is the investigation of a certain class of topological vector spaces over a fixed nonarchimedean field K. This is the class of locally convex vector spaces. The more traditional analytic point of view characterizes locally convex topologies as those vector space topologies which can be defined by a family of (nonarchimedean) seminorms. But the presence of the ring of integers o inside the field K allows for an equivalent algebraic point of view. A locally convex topology on a K-vector space V is a vector space topology defined by a class of o-submodules of V which are required to generate V as a vector space. In § §2 and 4 we thoroughly discuss these two concepts and their equivalence. Throughout the book we usually will present the theory from both angles. But sometimes there will be a certain bias towards the algebraic point of view.