Applications of direct and inverse limits in analysis

Abstract

In this dissertation, we use the categorical notions of direct and inverse limits to solve certain problems in analysis; in particular, in the field of vector lattices. Chapter 1 provides a general overview and motivation of the problems we will focus on. Specifically, these are a decomposition theorem for C(X) spaces that are order dual spaces, and the problem of existence of free objects in certain categories of locally convex structures. The connecting thread between these two disparate problems will be our extensive and fundamental use of direct and inverse limits in their solutions.

Chapter 2 deals with the first of these two problems. After settling some preliminaries, the first few sections of Chapter 2 develops the basic theory of direct and inverse limits in categories of vector lattices. This includes results on existence, permanence properties, as well as some examples. After this, we give some results on the duality between direct and inverse limits. In particular, we will show that the order (continuous) dual of a direct limit of vector lattices is an inverse limit of order (continuous) duals, and (under more strict conditions) the order (continuous) dual of an inverse limit of vector lattices is a direct limit of order (continuous) duals. The rest of Chapter 2 deals with applications of this duality theory in various contexts, among these will be our desired decomposition result for certain C(X) spaces, which is formulated in terms of an inverse limit.

Chapter 3 starts with some further preliminaries we need in order to define certain categories of algebraic structures, normed structures, and locally convex structures forming the setting of this chapter. After this, we cover some material from universal algebra to prove the existence of free objects in these algebraic categories. We use the existence of these algebraic free objects to expand upon the existing literature regarding certain `free objects' in categories of normed structures. As we shall detail below, these are not bona fide free objects in our sense of the term. Inverse limits re-enter the picture at this point: We will prove a general categorical result involving inverse limits that allows us to use our results for categories of normed structures to obtain genuine free objects in categories of locally convex structures. The abstract material in Chapter 3 will be interspersed with some concrete examples chosen from two particular cases. We conclude Chapter 3 by giving concrete descriptions of two free objects in certain categories of locally convex structures whose existence was proven using our general abstract methods.