Brown A., Pearcy C. An Introduction to Analysis

Springer, 1995. — 305 p. — ISBN: 978-1-4612-6901-4.As its title indicates, this book is intended to serve as a textbook for an introductory course in mathematical analysis. In preliminary form the
book has been used in this way at the University of Michigan, Indiana University, and Texas A&M University, and has proved serviceable. In addition to its primary purpose as a textbook for a formal course, however, it is the authors' hope that this book will also prove of value to readers
interested in studying mathematical analysis on their own. Indeed, we believe the wealth and variety of examples and exercises will be especially conducive to this end.
A word on prerequisites. With what mathematical background might a prospective reader hope to profit from the study of this book? Our conscious intent in writing it was to address the needs of a beginning graduate student in mathematics, or, to put matters slightly differently, a student
who has completed an undergraduate program with a mathematics major. On the other hand, the book is very largely self-contained and should
therefore be accessible to a lower classman whose interest in mathematical analysis has already been awakened.
The contents of the book may be briefly summarized. Chapters 1 through 3 constitute an overview of the preliminary material on which the rest of the book is built, viz., set theory, the number systems, and linear algebra. In no case do we imagine that this brief summary of material can serve as the reader's initial encounter with these ideas. Rather we have gathered together here the basic terminology and facts to be employed in all that follows. In particular, in Chapters 2 and 3 we introduce only material that is assumed to be already familiar to the reader, though perhaps in different form, and these two chapters may in most cases be treated quite lightly. Chapter 1, on the other hand, dealing with the rudiments of set
theory, acquaints the reader with inductive proofs based on the maximum principle in its various forms, and is deserving of more careful attention.
In Chapters 4 and 5 we present the essentials from the theory of transfinite numbers. This treatment, while concise, presents all of the ideas and
results that will actually be employed in the sequel, and is, in any case, fuller than is to be found in most other texts. In this connection we note
that the various number systems, formally introduced in Chapter 2, actually make a few brief cameo appearances in Chapter 1 as well. This minor
logical embarrassment could easily be averted, of course, but only at the cost of unwelcome circumlocutions.
Chapters 6 through 8 constitute the heart of the book. In them we explore in thoroughgoing fashion the structure of various metric spaces and the mappings defined on or taking values in such spaces. The topics and facts adduced are largely standard, though our choice of examples, problems, and manner of presentation may make some modest claim to freshness if not to novelty, but many of these lines of inquiry are pursued in greater detail than will be found in most other recent texts.
The final chapter (Chapter 9) consists of a treatment of general topology. In this chapter we equip the reader with the full panoply of topological
equipment needed for the transition from the world of classical analysis, set in metric spaces, to "modem" or "abstract" analysis, the realm of maximal ideal spaces, kernel-hull topologies, etc.
In formulating the sets of problems that follow each chapter we have followed current practice. Each problem, or part of a problem, is, in effect, a theorem to be proved, and it is our intention that the solutions should be written out with that in mind. Thus a problem posed as a simple yes-orno
question has for its proper solution not a simple yes-or-no answer, but rather an argument showing which is, in fact, correct. Similarly, a problem
posed as a statement of fact is really a disguised invitation to the reader to establish the validity of that fact. No conscious attempt was made to
grade the problems according to difficulty, but they are arranged in loosely chronological order, so that the first problems in each chapter relate to
the earlier parts of that chapter and subsequent problems to later parts.
Thus the earlier problems in anyone chapter do tum out, in general, to be somewhat easier than the later ones. (The problem sets are an integral
part of the text; an independent reader is advised to begin to look into the problem set at the end of a chapter as soon as he begins the perusal of
the chapter itself, just as he would do if assigned homework problems in a formal classroom setting.)