A collection of problems and solutions in real analysis based on the major textbook, Principles of Real Analysis (also by Aliprantis and Burkinshaw), Problems in Real Analysis is the ideal companion for senior science and engineering undergraduates and first-year graduate courses in real analysis. It is intended for use as an independent source, and is an invaluable tool for students who wish to develop a deep understanding and proficiency in the use of integration methods. Problems in Real Analysis teaches the basic methods of proof and problem-solving by presenting the complete solutions to over 600 problems that appear in Principles of Real Analysis, Third Edition . The problems are distributed in forty sections, and cover the entire spectrum of difficulty.
Author(s): Charalambos D. Aliprantis, Owen Burkinshaw
Edition: 2
Publisher: Academic Press
Year: 1999
Language: English
Pages: 412
Front cover......Page 1
Title page......Page 3
Copyright page......Page 4
CONTENTS......Page 5
Foreword......Page 7
1. Elementary Set Theory......Page 9
2. Countable and Uncountable Sets......Page 14
3. The Real Numbers......Page 19
4. Sequences of Real Numbers......Page 28
5. The Extended Real Numbers......Page 42
6. Metric Spaces......Page 53
7. Compactness in Metric Spaces......Page 62
8. Topological Spaces......Page 73
9. Continuous Real-Valued Functions......Page 81
10. Separation Properties of Continuous Functions......Page 100
11. The Stone-Weierstrass Approximation Theorem......Page 106
12. Semirings and Algebras of Sets......Page 115
13. Measures on Semirings......Page 120
14. Outer Measures and Measurable Sets......Page 124
15. The Outer Measure Generated by a Measure......Page 130
16. Measurable Functions......Page 141
17. Simple and Step Functions......Page 145
18. The Lebesgue Measure......Page 154
19. Convergence in Measure......Page 165
20. Abstract Measurability......Page 168
21. Upper Functions......Page 179
22. Integrable Functions......Page 182
23. The Riemann Integral as a Lebesgue Integral......Page 198
24. Applications of the Lebesgue Integral......Page 214
25. Approximating Integrable Functions......Page 228
26. Product Measures and Iterated Integrals......Page 232
27. Normed Spaces and Banach Spaces......Page 247
28. Operators between Banach Spaces......Page 253
29. Linear Functionals......Page 259
30. Banach Lattices......Page 267
31. $L_p$-Spaces......Page 279
32. Inner Product Spaces......Page 305
33. Hilbert Spaces......Page 318
34. Orthonormal Bases......Page 333
35. Fourier Analysis......Page 341
36. Signed Measures......Page 353
37. Comparing Measures and the Radon-Nikodym Theorem......Page 361
38. The Riesz Representation Theorem......Page 373
39. Differentiation and Integration......Page 387
40. The Change of Variables Formula......Page 403
Back cover......Page 412
