Foundations Of Mathematical Analysis
This book evolved from a one-year Advanced Calculus course that we have given during the last decade. Our audiences have included junior and senior majors and honors students, and, on occasion, gifted sophomores.
The material is logically self-contained; that is, all of our results are proved and are ultimately based on the axioms for the real numbers. We do not use results from other sources, except for a few results from linear algebra which are summarized in a brief appendix. Thus, theoretically, no prerequisites are necessary to understand this material. Realistically, the prerequisite is some mathematical maturity such as one might acquire by taking calculus and, perhaps, linear algebra.
Our intent is to teach students the tools of modern analysis as it relates to further study in mathematics, especially statistics, numerical analysis, differential equations, mathematical analysis, and functional analysis.
It is our belief that the key to a sound foundation for the study of analysis lies in an understanding of the limit concept. Thus, after initial chapters on sets and the real number system, we introduce the limit concept using numerical sequences and series (Chapters IV and V). This is followed by Chapter VI on the limit of a function. We then move to the general setting of metric spaces (Chapter VII). Chapter VIII is a review of differential calculus. Chapter IX gives a detailed introduction to the theory of Riemann-Stieltjes integration. We then turn to the study of sequences and series of functions (Chapters X and XI), Fourier series (Chapter XII), the Riesz representation theorem (Chapter XIII), and the Lebesgue integral (Chapter XIV). The first seven chapters could be used for a one-term course on the Concept of Limit.
Because we believe that an essential part of learning mathematics is doing mathematics, we have included over 750 exercises, some containing several parts, of varying degrees of difficulty. Hints and solutions to selected exercises, indicated by an asterisk, are given at the back of the book.
We would like to thank our colleagues, Dr. Rosalind Reichard, who taught this course from a preliminary version and gave us useful information, and Dr. Keith Rose, who read the manuscript and offered valuable criticism. Thanks also to our many students who studied this material and offered suggestions, and especially Mr. James Africh, who worked nearly every exercise and made many helpful comments. Our thanks also go to the secretarial staff at the University of Victoria, who over the years typed various versions of the manuscript. Of course, we assume joint responsibility for the book’s strengths and weaknesses, and we welcome comment.
W. E. Pfaffenberger