Principles of Real Analysis, Third Edition

✍ Scribed by Charalambos D. Aliprantis

• Publisher: Academic Press
• Year: 1998
• Tongue: English
• Leaves: 426
• Edition: 3
• Category: Library

✦ Synopsis

With the success of its previous editions, Principles of Real Analysis, Third Edition, continues to introduce students to the fundamentals of the theory of measure and functional analysis. In this thorough update, the authors have included a new chapter on Hilbert spaces as well as integrating over 150 new exercises throughout. The new edition covers the basic theory of integration in a clear, well-organized manner, using an imaginative and highly practical synthesis of the "Daniell Method" and the measure theoretic approach. Students will be challenged by the more than 600 exercises contained in the book. Topics are illustrated by many varied examples, and they provide clear connections between real analysis and functional analysis. Gives a unique presentation of integration theory Over 150 new exercises integrated throughout the text Presents a new chapter on Hilbert Spaces Provides a rigorous introduction to measure theory Illustrated with new and varied examples in each chapter Introduces topological ideas in a friendly manner Offers a clear connection between real analysis and functional analysis Includes brief biographies of mathematicians

✦ Table of Contents

CONTENTS......Page 5
PREFACE......Page 7
1. Elementary Set Theory......Page 11
2. Countable and Uncountable Sets......Page 19
3. The Real Numbers......Page 24
4. Sequences of Real Numbers......Page 32
5. The Extended Real Numbers......Page 39
6. Metric Spaces......Page 44
7. Compactness in Metric Spaces......Page 58
8. Topological Spaces......Page 67
9. Continuous Real-Valued Functions......Page 76
10. Separation Properties of Continuous Functions......Page 90
11. The Stone-Weierstrass Approximation Theorem......Page 97
12. Semirings and Algebras of Sets......Page 103
13. Measures on Semirings......Page 108
14. Outer Measures and Measurable Sets......Page 113
15. The Outer Measure Generated by a Measure......Page 120
16. Measurable Functions......Page 130
17. Simple and Step Functions......Page 136
18. The Lebesgue Measure......Page 143
19. Convergence in Measure......Page 156
20. Abstract Measurability......Page 159
21. Upper Functions......Page 171
22. Integrable Functions......Page 176
23. The Riemann Integral as a Lebesgue Integral......Page 187
24. Applications of the Lebesgue Integral......Page 200
25. Approximating Integrable Functions......Page 211
26. Product Measures and Iterated Integrals......Page 214
27. Normed Spaces and Banach Spaces......Page 227
28. Operators Between Banach Spaces......Page 234
29. Linear Functionals......Page 245
30. Banach Lattices......Page 252
31. L_p-Spaces......Page 264
6. HILBERT SPACES......Page 285
32. Inner Product Spaces......Page 286
33. Hilbert Spaces......Page 298
34. Orthonormal Bases......Page 308
35. Fourier Analysis......Page 317
7. SPECIAL TOPICS IN INTEGRATION......Page 335
36. Signed Measures......Page 336
37. Comparing Measures and the Radon-Nikodym Theorem......Page 348
38. The Riesz Representation Theorem......Page 362
39. Differentiation and Integration......Page 376
40. The Change of Variables Formula......Page 395
BIBLIOGRAPHY......Page 409
LIST OF SYMBOLS......Page 411
INDEX......Page 413