Analysis
An Introduction
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Short description/annotationThis book is an extensive introductory text to mathematical analysis for graduate students and advanced undergraduates, complete with 500 exercises and numerous examples.Main descriptionThis self-contained text, suitable for advanced undergraduates, provides an extensive introduction to mathematical analysis, from the fundamentals to more advanced material. It begins with the properties of the real numbers and continues with a rigorous treatment of sequences, series, metric spaces, and calculus in one variable. Further subjects include Lebesgue measure and integrati...
Short description/annotation
This book is an extensive introductory text to mathematical analysis for graduate students and advanced undergraduates, complete with 500 exercises and numerous examples.
Main description
This self-contained text, suitable for advanced undergraduates, provides an extensive introduction to mathematical analysis, from the fundamentals to more advanced material. It begins with the properties of the real numbers and continues with a rigorous treatment of sequences, series, metric spaces, and calculus in one variable. Further subjects include Lebesgue measure and integration on the line, Fourier analysis, and differential equations. In addition to this core material, the book includes a number of interesting applications of the subject matter to areas both within and outside the field of mathematics. The aim throughout is to strike a balance between being too austere or too sketchy, and being so detailed as to obscure the essential ideas. A large number of examples and 500 exercises allow the reader to test understanding, practise mathematical exposition and provide a window into further topics.
Table of contents:
1. Introduction; 2. The real and complex numbers; 3. Real and complex sequences; 4. Series; 5. Power series; 6. Metric spaces; 7. Continuous functions; 8. Calculus; 9. Some special functions; 10. Lebesgue measure on the line; 11. Lebesgue integration on the line; 12. Function spaces; 13. Fourier series; 14. Applications of Fourier series; 15. Ordinary differential equations; Appendix: the Banach-Tarski paradox; Hints for some exercises.
This book is an extensive introductory text to mathematical analysis for graduate students and advanced undergraduates, complete with 500 exercises and numerous examples.
Main description
This self-contained text, suitable for advanced undergraduates, provides an extensive introduction to mathematical analysis, from the fundamentals to more advanced material. It begins with the properties of the real numbers and continues with a rigorous treatment of sequences, series, metric spaces, and calculus in one variable. Further subjects include Lebesgue measure and integration on the line, Fourier analysis, and differential equations. In addition to this core material, the book includes a number of interesting applications of the subject matter to areas both within and outside the field of mathematics. The aim throughout is to strike a balance between being too austere or too sketchy, and being so detailed as to obscure the essential ideas. A large number of examples and 500 exercises allow the reader to test understanding, practise mathematical exposition and provide a window into further topics.
Table of contents:
1. Introduction; 2. The real and complex numbers; 3. Real and complex sequences; 4. Series; 5. Power series; 6. Metric spaces; 7. Continuous functions; 8. Calculus; 9. Some special functions; 10. Lebesgue measure on the line; 11. Lebesgue integration on the line; 12. Function spaces; 13. Fourier series; 14. Applications of Fourier series; 15. Ordinary differential equations; Appendix: the Banach-Tarski paradox; Hints for some exercises.
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