Book:Charalambos D. Aliprantis/Principles of Real Analysis/Third Edition

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Charalambos D. Aliprantis and Owen Burkinshaw: Principles of Real Analysis (3rd Edition)

Published $1998$, Academic Press

ISBN 978-0120502578.


Subject Matter


Contents

Preface
Chapter 1. Fundamentals of Real Analysis
1. Elementary Set Theory I
2. Countable and Uncountable Sets
3. The Real Numbers
4. Sequences of Real Numbers
5. The Extended Real Numbers
6. Metric Spaces
7. Compactness in Metric Spaces
Chapter 2. Topology and Continuity
8. Topological Spaces
9. Continuous Real-Valued Functions
10. Separation Properties of Continuous Functions
11. The Stone-Weierstrass Approximation Theorem
Chapter 3. The Theory of Measure
12. Semirings and Algebras of Sets
13. Measures on Semirings
14. Outer Measures and Measurable Sets
15. The Outer Measure Generated by a Measure
16. Measurable Functions
17. Simple and Step Functions
18. The Lebesgue Measure
19. Convergence in Measure
20. Abstract Measurability
Chapter 4. The Lebesgue Integral
21. Upper Functions
22. Integrable Functions
23. The Riemann Integral as a Lebesgue Integral
24. Applications of the Lebesgue Integral
25. Approximating Integrable Functions
26. Product Measures and Iterated Integrals
Chapter 5. Normed Spaces and $L_p$-spaces
27. Normed Spaces and Banach Spaces
28. Operators Between Banach Spaces
29. Linear Functionals
30. Banach Lattices
31. $L_p$-Spaces
Chapter 6. Hilbert Spaces
32. Inner Product Spaces
33. Hilbert Spaces
34. Orthonormal Bases
35. Fourier Analysis
Chapter 7. Special Topics in Integration
36. Signed Measures
37. Comparing Measures and the Radmi-Nikodym Theorem
38. The Riesz Representation Theorem
39. Differentiation and Integration
40. The Change of Variables Formula
Bibliography
List of Symbols
Index