Foundations of Mathematical Analysis by Johnsonbaugh, Richard -- Read -- Imperial Library of Trantor

Index

Cover Title Page Copyright Preface Preface to the Dover Edition Contents I Sets and Functions

1. Sets 2. Functions

II The Real Number System

3. The Algebraic Axioms of the Real Numbers 4. The Order Axiom of the Real Numbers 5. The Least-Upper-Bound Axiom 6. The Set of Positive Integers 7. Integers, Rationals, and Exponents

III Set Equivalence

8. Definitions and Examples 9. Countable and Uncountable Sets

IV Sequences of Real Numbers

10. Limit of a Sequence 11. Subsequences 12. The Algebra of Limits 13. Bounded Sequences 14. Further Limit Theorems 15. Divergent Sequences 16. Monotone Sequences and the Number e 17. Real Exponents 18. The Bolzano-Weierstrass Theorem 19. The Cauchy Condition 20. The lim sup and lim inf of Bounded Sequences 21. The lim sup and lim inf of Unbounded Sequences

V Infinite Series

22. The Sum of an Infinite Series 23. Algebraic Operations on Series 24. Series with Nonnegative Terms 25. The Alternating Series Test 26. Absolute Convergence 27. Power Series 28. Conditional Convergence 29. Double Series and Applications

VI Limits of Real-Valued Functions and Continuous Functions on the Real Line

30. Definition of the Limit of a Function 31. Limit Theorems for Functions 32. One-Sided and Infinite Limits 33. Continuity 34. The Heine-Borel Theorem and a Consequence for Continuous Functions

VII Metric Spaces

35. The Distance Function 36. Rn, l2, and the Cauchy-Schwarz Inequality 37. Sequences in Metric Spaces 38. Closed Sets 39. Open Sets 40. Continuous Functions on Metric Spaces 41. The Relative Metric 42. Compact Metric Spaces 43. The Bolzano-Weierstrass Characterization of a Compact Metric Space 44. Continuous Functions on Compact Metric Spaces 45. Connected Metric Spaces 46. Complete Metric Spaces 47. Baire Category Theorem

VIII Differential Calculus of the Real Line

48. Basic Definitions and Theorems 49. Mean-Value Theorems and L‘Hospital’s Rule 50. Taylor's Theorem

IX The Riemann-Stieltjes Integral

X Sequences and Series of Functions

60. Pointwise Convergence and Uniform Convergence 61. Integration and Differentiation of Uniformly Convergent Sequences 62. Series of Functions 63. Applications to Power Series 64. Abel's Limit Theorems 65. Summability Methods and Tauberian Theorems

XI Transcendental Functions

66. The Exponential Function 67. The Natural Logarithm Function 68. The Trigonometric Functions

XII Inner Product Spaces and Fourier Series

69. Normed Linear Spaces 70. The Inner Product Space R3 71. Inner Product Spaces 72. Orthogonal Sets in Inner Product Spaces 73. Periodic Functions 74. Fourier Series: Definition and Examples 75. Orthonormal Expansions in Inner Product Spaces 76. Pointwise Convergence of Fourier Series in R[a, a + 2π] 77. Cesàro Summability of Fourier Series 78. Fourier Series in R[a, a + 2π] 79. A Tauberian Theorem and an Application to Fourier Series

XIII Normed Linear Spaces and the Riesz Representation Theorem

XIV The Lebesgue Integral

85. The Extended Real Line 86. σ-Algebras and Positive Measures 87. Measurable Functions 88. Integration on Positive Measure Spaces 89. Lebesgue Measure on R 90. Lebesgue Measure on [a, b] 91. The Hilbert Spaces L2(X, M, μ)

Appendix: Vector Spaces References Hints to Selected Exercises Index Errata