Real and complex analysis /
Rudin, Walter,
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
9780071002769
Mathematical analysis.
515 / R9161
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
9780071002769
Mathematical analysis.
515 / R9161
