Principles of Mathematical Analysis

Cover......Page 1
Title......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 10
Introduction......Page 12
Ordered Sets......Page 14
Fields......Page 16
The Real Field......Page 19
The Extended Real Number System......Page 22
The Complex Field......Page 23
Euclidean Spaces......Page 27
Appendix......Page 28
Exercises......Page 32
Finite, Countable, and Uncountable Sets......Page 35
Metric Spaces......Page 41
Compact Sets......Page 47
Perfect Sets......Page 52
Connected Sets......Page 53
Exercises......Page 54
Convergent Sequences......Page 58
Subsequences......Page 62
Cauchy Sequences......Page 63
Upper and Lower Limits......Page 66
Some Special Sequences......Page 68
Series......Page 69
Series of Nonnegative Terms......Page 72
The Number e......Page 74
The Root and Ratio Tests......Page 76
Power Series......Page 80
Summation by Parts......Page 81
Absolute Convergence......Page 82
Addition and Multiplication of Series......Page 83
Rearrangements......Page 86
Exercises......Page 89
Limits of Functions......Page 94
Continuous Functions......Page 96
Continuity and Compactness......Page 100
Continuity and Connectedness......Page 104
Discontinuities......Page 105
Monotonic Functions......Page 106
Infinite Limits and Limits at Infinity......Page 108
Exercises......Page 109
The Derivative of a Real Function......Page 114
Mean Value Theorems......Page 208
The Continuity of Derivatives......Page 119
L\'Hospital\'s Rule......Page 120
Taylor\'s Theorem......Page 121
Differentiation of Vector-valued Functions......Page 122
Exercises......Page 125
Definition and Existence of the Integral......Page 131
Properties of the Integral......Page 139
Integration and Differentiation......Page 144
Integration of Vector-valued Functions......Page 146
Rectifiable Curves......Page 147
Exercises......Page 149
Discussion of Main Problem......Page 154
Uniform Convergence......Page 158
Uniform Convergence and Continuity......Page 160
Uniform Convergence and Integration......Page 162
Uniform Convergence and Different iat ion......Page 163
Equicontinuous Families of Functions......Page 165
The Stone-Weierstrass Theorem......Page 170
Exercises......Page 176
Power Series......Page 183
The Exponential and Logarithmic Functions......Page 189
The Trigonometric Functions......Page 193
The Algebraic Completeness of the Complex Field......Page 195
Fourier Series......Page 196
The Gamma Function......Page 203
Exercises......Page 207
Linear Transformations......Page 215
Differentiation......Page 222
The Contraction Principle......Page 231
The Inverse Function Theorem......Page 232
The Implicit Function Theorem......Page 234
The Rank Theorem......Page 239
Determinants......Page 242
Derivatives of Higher Order......Page 246
Differentiation of Integrals......Page 247
Exercises......Page 250
Integration......Page 256
Primitive Mappings......Page 259
Partitions of Unity......Page 262
Change of Variables......Page 263
Differential Forms......Page 264
Simplexes and Chains......Page 267
Stokes\' Theorem......Page 284
Closed Forms and Exact Forms......Page 286
Vector Analysis......Page 291
Exercises......Page 299
Set Functions......Page 311
Construction of the Lebesgue Measure......Page 313
Measurable Functions......Page 321
Simple Functions......Page 324
Integration......Page 325
Comparison with the Riemann Integral......Page 333
Functions of Class L^2......Page 336
Exercises......Page 343
Bibliography......Page 346
List of Special Symbols......Page 348
Index......Page 350