Real and complex analysis /
Walter Rudin.
- 3rd ed.
- New York : McGraw-Hill, c1987.
- xiv, 416 p. 24 cm.
Cover title: Real & complex analysis. Includes index.
1. Preface 2. Prologue: the exponential function 3. Chapter 1: abstract integration 4. Set-theoretic notations and terminology 5. The concept of measurability 6. Simple functions 7. Elementary properties of measures 8. Arithmetic in [0, infinity] 9. Integration of positive functions 10. Integration of complex functions 11. The role played by sets of measure zero 12. Exercises 13. Chapter 2: positive borel measures 14. Vector spaces 15. Topological preliminaries 16. The riesz representation theorem 17. Regularity properties of borel measures 18. Lebesgue measure 19. Continuity properties of measurable functions 20. Exercises 21. Chapter 3: l^p-spaces 22. Convex functions and inequalities 23. The l^p-spaces 24. Approximation by continuous functions 25. Exercises 26. Chapter 4: elementary hilbert space theory 27. Inner products and linear functionals 28. Orthonormal sets 29. Trigonometric series 30. Exercises 31. Chapter 5: examples of banach space techniques 32. Banach spaces 33. Consequences of baire's theorem 34. Fourier series of continuous functions 35. Fourier coefficients of l¹-functions 36. The hahn-banach theorem 37. An abstract approach to the poisson integral 38. Exercises 39. Chapter 6: complex measures 40. Total variation 41. Absolute continuity 42. Consequences of the radon-nikodym theorem 43. Bounded linear functionals on l^p 44. The riesz representation theorem 45. Exercises 46. Chapter 7: differentiation 47. Derivatives of measures 48. The fundamental theorem of calculus 49. Differentiable transformations 50. Exercises 51. Chapter 8: integration on product spaces 52. Measurability on cartesian products 53. Product measures 54. The fubini theorem 55. Completion of product measures 56. Convolutions 57. Distribution functions 58. Exercises 59. Chapter 9: fourier transforms 60. Formal properties 61. The inversion theorem 62. The plancherel theorem 63. The banach algebra l¹ 64. Exercises 65. Chapter 10: elementary properties of holomorphic functions 66. Complex differentiation 67. Integration over paths 68. The local cauchy theorem 69. The power series representation 70. The open mapping theorem 71. The global cauchy theorem 72. The calculus of residues 73. Exercises 74. Chapter 11: harmonic functions 75. The cauchy-riemann equations 76. The poisson integral 77. The mean value property 78. Boundary behavior of poisson integrals 79. Representation theorems 80. Exercises 81. Chapter 12: the maximum modulus principle 82. Introduction 83. The schwarz lemma 84. The phragmen-lindelöf method 85. An interpolation theorem 86. A converse of the maximum modulus theorem 87. Exercises 88. Chapter 13: approximation by rational functions 89. Preparation 90. Runge's theorem 91. The mittag-leffler theorem 92. Simply connected regions 93. Exercises 94. Chapter 14: conformal mapping 95. Preservation of angles 96. Linear fractional transformations 97. Normal families 98. The riemann mapping theorem 99. The class µ 100. Continuity at the boundary 101. Conformal mapping of an annulus 102. Exercises 103. Chapter 15: zeros of holomorphic functions 104. Infinite products 105. The weierstrass factorization theorem 106. An interpolation problem 107. Jensen's formula 108. Blaschke products 109. The müntz-szas theorem 110. Exercises 111. Chapter 16: analytic continuation 112. Regular points and singular points 113. Continuation along curves 114. The monodromy theorem 115. Construction of a modular function 116. The picard theorem 117. Exercises 118. Chapter 17: h^p-spaces 119. Subharmonic functions 120. The spaces h^p and n 121. The theorem of f. And m. Riesz 122. Factorization theorems 123. The shift operator 124. Conjugate functions 125. Exercises 126. Chapter 18: elementary theory of banach algebras 127. Introduction 128. The invertible elements 129. Ideals and homomorphisms 130. Applications 131. Exercises 132. Chapter 19: holomorphic fourier transforms 133. Introduction 134. Two theorems of paley and wiener 135. Quasi-analytic classes 136. The denjoy-carleman theorem 137. Exercises 138. Chapter 20: uniform approximation by polynomials 139. Introduction 140. Some lemmas 141. Mergelyan's theorem 142. Exercises 143. Appendix: hausdorff's maximality theorem 144. Notes and comments 145. Bibliography 146. List of special symbols 147. Index