This second edition is a corrected, revised, and reprinted version of our original textbook. We are particularly grateful to readers who have sent in suggestions for corrections. Among them we owe a huge debt to R. B. Burckel (Kansas State University). Many of his corrections and suggestions are incorporated in this new edition. Thanks too to Keith Yates (Manchester Metropolitan University) who, while working on some of the more difficult problems, found some further errors.
In teaching first courses in real analysis over the years, we have found increasingly that the classes form rather heterogeneous groups. It is no longer true that most of the students are first-year graduate students in mathematics, presenting more or less common backgrounds for the course. Indeed, nowadays we find diverse backgrounds and diverse objectives among students in such classes. Some students are undergraduates, others are more advanced. Many students are in other departments, such as statistics or engineering. Some students are seeking terminal master’s degrees; others wish to become research mathematicians, not necessarily in analysis.
We have tried to write a book that is suitable for students with minimal backgrounds, one that does not presuppose that most students will eventually specialize in analysis. We have pursued two goals. First, we would like all students to have an opportunity to obtain an appreciation of the tools, methods, and history of the subject and a sense of how the various topics we cover develop naturally. Our second objective is to provide those who will study analysis further with the necessary background in measure, integration, differentiation, metric space theory, and functional analysis. To meet our first goal, we do several things. We provide a certain amount of historical perspective that may enable a reader to see why a theory was needed and sometimes, why the researchers of the time had difficulty obtaining the “right” theory. We try to motivate topics before we develop them and try to motivate the proofs of some of the important theorems that students often find difficult. We usually avoid proofs that may appear “magical” to students in favor of more revealing proofs that may be a bit longer. We describe the interplay of various subjects—measure, variation, integration, and differentiation. Finally, we indicate applications of abstract theorems such as the contraction mapping principle, the Baire category theorem, Ascoli’s theorem, Hahn-Banach theorem, and the open mapping theorem, to concrete settings of various sorts.
We consider the exercise sections an important part of the book. Some of the exercises do no more than ask the reader to complete a proof given in the text, or to prove an easy result that we merely state. Others involve simple applications of the theorems. A number are more ambitious. Some of these exercises extend the theory that we developed or present some related material. Others provide examples that we believe are interesting and revealing, but may not be well known. In general, the problems at the ends of the chapters are more substantial. A few of these problems can form the basis of projects for further study.