Richard Johnsonbaugh - Foundations of Mathematical Analysis

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 (). 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. Richard Johnsonbaugh
W. E. PfaffenbergerPreface to the Dover EditionFoundations of Mathematical Analysis covers real analysisfrom nave set theory and the axioms for the real numbers to the Lebesgue integral, with sequences and series, metric spaces, the Riemann-Stieltjes integral, inner product spaces, Fourier series, Tauberian theorems, the Riesz representation theorem, and a brief discussion of Hilbert spaces in between. The book evolved from a one-year Advanced Calculus course that both authors taught during the late 1960s and throughout the 1970s. The audience included junior and senior majors and honors students, and, on occasion, gifted sophomores.

Professor Pfaffenberger has continued to teach this course. (Professor Johnsonbaugh moved to computer science.) The intervening years have confirmed the importance of real analysis. Analysis is a core subject in mathematics and is a prerequisite for further study in mathematics. Analysis is also fundamental to many related fields such as statistics. Several of Professor Pfaffenberger's students have completed doctorates at distinguished institutions (e.g., Princeton, Harvard, Berkeley, Cambridge), and many have specialized in analysis. 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 in the back of the book. We will maintain a World Wide Web site for this reprint that contains additional problems, errata, and other supplementary material. About This Book 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 that are summarized in a brief appendix. Thus, logically, 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.

It is our belief that understanding the limit concept is the key to a sound foundation for the study of analysis. Thus, after initial chapters on sets and the real number system (). In ). The first seven chapters could be used for a one-term course on the Concept of Limit. After a review of differential calculus (). discusses the exponential, logarithm, and trigonometric functions. discusses the exponential, logarithm, and trigonometric functions.

The exponential, sine, and cosine functions are defined by power series, after which their standard properties are derived. The other trigonometric functions are defined in terms of the sine and cosine functions, and the logarithm function is defined as the inverse of the exponential function. Inner product spaces and Fourier series are the topics for ). ), where C[a, b] is the set of continuous functions on [a, b]: If T is a continuous linear functional on C[a, b], then there exists a function of bounded variation on [a, b] such that Furthermore, the norm of T is equal to the total variation of on [a, b]. The last chapter (). The book concludes with an appendix on vector spaces, a list of references, and hints to selected exercises.

The appendix summarizes some of the key definitions and theorems concerning vector spaces that are used in the book. Examples The book contains nearly 100 worked examples. These examples clarify the theory, show students how to develop proofs, demonstrate applications of the theory, elucidate proofs, and help motivate the material. World Wide Web Site The World Wide Web site http://condor.depaul.edu/rjohnson contains Expanded explanations of particular topics and further information on analysis. Additional exercises. Acknowledgments We thank our former colleagues Rosalind Reichard and Keith Rose, who offered valuable comments on early drafts of the book. Acknowledgments We thank our former colleagues Rosalind Reichard and Keith Rose, who offered valuable comments on early drafts of the book.

Thanks also to our many students, who studied this material and contributed suggestionsin particular, we thank James Africh, who worked most of the exercises. Our thanks also go to the secretarial staff at the University of Victoria, who typed many drafts of the book. In particular, we are grateful to Yvonne Leeming for typing the first camera-ready version of the book. Finally, we thank John W. Grafton, Senior Reprint Editor, Dover Publications, for proposing this edition of the book and for his assistance in producing it.