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The book targets undergraduate and postgraduate mathematics students and helps them develop a deep understanding of mathematical analysis. Designed as a first course in real analysis, it helps students learn how abstract mathematical analysis solves mathematical problems that relate to the real world. As well as providing a valuable source of inspiration for contemporary research in mathematics, the book helps students read, understand and construct mathematical proofs, develop their problem-solving abilities and comprehend the importance and frontiers of computer facilities and much more.
It offers comprehensive material for both seminars and independent study for readers with a basic knowledge of calculus and linear algebra. The first nine chapters followed by the appendix on the Stieltjes integral are recommended for graduate students studying probability and statistics, while the first eight chapters followed by the appendix on dynamical systems will be of use to studentsof biology and environmental sciences. Chapter 10 and the appendixes are of interest to those pursuing further studies at specialized advanced levels. Exercises at the end of each section, as well as commentaries at the end of each chapter, further aid readers' understanding. The ultimate goal of the book is to raise awareness of the fine architecture of analysis and its relationship with the other fields of mathematics.
Inhaltsverzeichnis
Preface. - Chapter 1. The Real Numbers. - Chapter 2. Limits of Real Sequences. - Chapter 3. The Euclidean Spaces RP and C. - Chapter 4. Numerical Series. - Chapter 5. Metric and Topology. - Chapter 6. Continuous Functions. - Chapter 7. Elementary Functions. - Chapter 8. Differential Calculus on R. - Chapter 9. The Riemann Integral. - Chapter 10. Improper Riemann Integrals. - Chapter 11. The Theory of Lebesgue Integral. - Chapter 12. Fourier Series. - Appendices.
The authors of this book intend to familiarize the reader with the basic concepts, principles, and methods of analysis and to ease access to more advanced topics. Every section of the book ends with exercises and every chapter of the book ends with a section of notes and remarks that provides historical information and supplementary material devoted to a better understanding of the present state of real analysis. I hope that this book achieves its mission. (Mansoor Saburov, Mathematical Reviews, January, 2016)
This book is a classical textbook on the theory of one variable real functions. The book is carefully written, with rigorous proofs and a sufficient number of problems, both solved and unsolved. The book goes under the surface of the topic and it is aimed in the first place to students of science . (Vladimír Janiš, zbMATH 1317. 26001, 2015)
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