Real Analysis
Cambridge University Press, 8/21/2008
EAN 9780521497565, ISBN10: 0521497566
Paperback, 416 pages, 25.4 x 17.8 x 2.4 cm
Language: English
This is a course in real analysis directed at advanced undergraduates and beginning graduate students in mathematics and related fields. Presupposing only a modest background in real analysis or advanced calculus, the book offers something to specialists and non-specialists. The course consists of three major topics: metric and normed linear spaces, function spaces, and Lebesgue measure and integration on the line. In an informal style, the author gives motivation and overview of new ideas, while supplying full details and proofs. He includes historical commentary, recommends articles for specialists and non-specialists, and provides exercises and suggestions for further study. This text for a first graduate course in real analysis was written to accommodate the heterogeneous audiences found at the masters level: students interested in pure and applied mathematics, statistics, education, engineering, and economics.
Preface
Part I. Metric Spaces
1. Calculus review
2. Countable and uncountable sets
3. Metrics and norms
4. Open sets and closed sets
5. Continuity
6. Connected sets
7. Completeness
8. Compactness
9. Category
Part II. Function Spaces
10. Sequences of functions
11. The space of continuous functions
12. The Stone-Weierstrass theorem
13. Functions of bounded variation
14. The Riemann-Stieltjes integral
15. Fourier series
Part III. Lebesgue Measure and Integration
16. Lebesgue measure
17. Measurable functions
18. The Lebesgue integral
19. Additional topics
20. Differentiation
References
Index.