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Introduction to real analysis. An educational approach. (English) Zbl 1183.00011
Hoboken, NJ: John Wiley & Sons (ISBN 978-0-470-37136-7/hbk). xv, 262 p. (2009).
This book is an interesting one about real analysis, a fascinating elegant area containing many deep results that are important throughout mathematics. The first chapter is a quick overview of the standard core of a first-year college calculus with examples and problems illustrating the topics, pausing to examine problems illustrating links and potentials difficulties. The problems illustrate both examples and counterexamples and lead to further thought. The author presents the study of real analysis in Chapter 2 starting with basic properties of the real numbers and going through limits to integration to series of functions in a natural progression, focusing on the results and proofs that help to lead to a deeper understanding of basics. Historically, the author follows the growth of real analysis from Cauchy to Riemann. Chapter 3 is an introduction to Lebesgue theory, analysis from a very advanced viewpoint. Understanding the basics of Lebesgue’s approach to integration and measure provides a superior foundation for elementary real analysis and offers a view toward more advanced topics. The last chapter is a selection of special topics that lead to other areas of study. These topics form the germs of excellent projects. The appendices serve as references. Appendix A lists the definitions and theorems of elementary analysis. Appendix B offers a very brief timeline placing Newton and Leibniz in perspective and Appendix C is a collection of projects. Appendix C also has pointers to other sources of projects, both for real analysis and for calculus. The historical development is interesting: history helps to see the connections among the themes of analysis. The book helps the student to place the mathematics and to better understand the imperative of developments.
MSC:
00A35 Methodology of mathematics
26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
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