In this prologue we set the stage by briefly discussing some points of notation and terminology and a few facts from set theory that will be used throughout the book.

We set

$\mathbb{N}$– the set of positive integers,

$\mathbb{Z}$= the set of integers,

$\mathbb{R}$= the set of real numbers,

$\mathbb{C}$– the set of complex numbers.

We often enlarge the real number system by adjoining two “elements at infinity,”$\infty $(also called$ + \infty $for emphasis) and$ - \infty $. In the extended system$\mathbb{R} \cup \{ \pm \infty \} = [ - \infty ,\infty ]$, every setEhas a least upper bound orsupremumand a greatest lower bound orinfimum,...

The subject of this chapter ispoint-set topologyorgeneral topology, the abstract mathematical framework for the study of limits, continuity, and the related geometric properties of sets.

The early years of the twentieth century witnessed a great increase in the level of abstraction and generality in mathematical thinking. In particular, mathematicians at that time developed theories that provide a very general setting for studying the circle of ideas related to limits and continuity, which previously had been considered in the context of subsets of Euclidean space or functions of one or several real or complex variables.

The most straightforward...

The theory of measure of subsets of Euclidean space (length, area, volume, and their analogues in higher dimensions) and the closely related theory of integration of functions on Euclidean space have a very long history. Much of the modern theory, however, does not depend on the particular features of the geometry of Euclidean space. It can be developed in a much more general setting with no additional effort, and in this more general form it yields results that can be applied in many additional situations.

This abstract theory is the subject of the present chapter; the methods for constructing interesting...

In this chapter we begin by presenting a general scheme for constructing measures. We then use it to construct Lebesgue measure and related measures on Euclidean space, and we analyze these measures and their associated integrals in some detail. We conclude with a discussion of regular Borel measures and integrals on locally compact Hausdorff spaces.

The construction of nontrivial examples of measures is not easy. To motivate the ideas, let us consider the elementary notion of area for regions in the plane${\mathbb{R}^2}$that is defined in terms of grids of rectangles. We first define the area of rectangle (the...

Functional analysis — the meeting ground of analysis and linear algebra, mostly in an infinite-dimensional setting — is a vast subject, and this brief account does no more than scratch the surface. Our object is simply to introduce some basic concepts that are of wide utility and a few fundamental theorems: just enough to support the material in the last two chapters of this book. For those who want to learn more there are many books available; Reed and Simon [14] and Rudin [18] are among the best.

Letxbe a vector space over the field$\mathbb{F}$, where$\mathbb{F}$is either...

In this chapter we study some of the spaces of functions that are of fundamental importance in modern analysis.

Let$(X,M,\mu )$be a measure space. We recall that${L^1}(\mu )$is the space of allμ-integrable complex-valued functions onX. For$0 < P < \infty $, we define the space${L^p}(\mu )$(also denoted by${L^p}(X)$or simply${L^p}$whenμis understood) to be the set of all measurable complex-valued functionsfonXsuch that${\left| f \right|^p} \in {L^1}(\mu )$. Thus, with the notation

(5.1)${\left\| f \right\|_p} = {\left[ {\int {{{\left| f \right|}^p}d\mu } } \right]^{1/p}}$,

${L^p}(\mu )$is the space of all measurable functions onXsuch that${\left\| f \right\|_p} < \infty $. As in the case of${L^1}(\mu )$, two functions are...

In this chapter we present a few basic applications of the abstract ideas and results from the preceding chapters in the concrete setting of the analysis of functions of one or several real variables. There is much more to be said; we are merely scratching the surface of a vast subject that has undergone a vigorous development in the last century. Some references for more extensive treatments include Dym and McKean [2], Strichartz [21], and Stein [20].

We begin with a few matters of notation. First, we shall denote the integral of a functionfon${\mathbb{R}^n}$with respect to...