Preface
Chapter 1 The Real and Complex Number Systems 1
Introduction 1
Ordered Sets 3
Fields 5
The Real Field 8
The Extended Real Number System 11
The Complex Field 12
Euclidean Spaces 16
Appendix 17
Exercises 21
Chapter 2 Basic Topology 24
Finite, Countable, and Uncountable Sets 24
Metric Spaces 30
Compact Sets 36
Perfect Sets 41 Connected Sets 42
Exercises 43
Chapter 3 Numerical Sequences and Series 47
Convergent Sequences 47
Subsequences 51
Cauchy Sequences 52
Upper and Lower Limits 55
Some Special Sequences 57
Series 58
Series of Nonnegative Terms 61
The Number e 63
The Root and Ratio Tests 65
Power Series 69
Summation by Parts 70
Absolute Convergence 71
Addition and Multiplication of Series 72
Rearrangements 75
Exercises 78
Chapter 4 Continuity 83
Limits of Functions 83
Continuous Functions 85
Continuity and Compactness 89
Continuity and Connectedness 93
Discontinuities 94
Monotonic Functions 95
Infinite Limits and Limits at Infinity 97
Exercises 98
Chapter 5 Differentiation 103
The Derivative of a Real Function 103
Mean Value Theorems 107
The Continuity of Derivatives 108
L'Hospital's Rule 109
Derivatives of Higher Order 110
Taylor’s Theorem 110
Differentiation of Vector-valued Functions 111
Exercises 114
　　　　Chapter 6 The Riemann-Stieltjes Integral 120
　　　　Definition and Existence of the Integral 120
　　　　Properties of the Integral 128
　　　　Integration and Differentiation 133
　　　　Integration of Vector-valued Functions 135
　　　　Rectifiable Curves 136
　　　　Exercises 138
　　　　Chapter 7 Sequences and Series of Functions, 143
　　　　Discussion of Main Problem 143
　　　　Uniform Convergence 147
　　　　Uniform Convergence and Continuity 149
　　　　Uniform Convergence and Integration 151
　　　　Uniform Convergence and Differentiation 152
　　　　Equicontinuous Families of Functions 154
　　　　The Stone-Weierstrass Theorem 159
　　　　Exercises 165
　　　　Chapter 8 Some Special Functions 172
　　　　Power Series 172
　　　　The Exponential and Logarithmic Functions 178
　　　　The Trigonometric Functions 182
　　　　The Algebraic Completeness of the Complex Field 184
　　　　Fourier Series 185
　　　　The Gamma Function 192
　　　　Exercises 196
　　　　Chapter 9 Functions of Several Variables 204
　　　　Linear Transformations 204
　　　　Differentiation 211
　　　　The Contraction Principle 220
　　　　The Inverse Function Theorem 221
　　　　The Implicit Function Theorem 223
　　　　The Rank Theorem 228
　　　　Determinants 231
　　　　Derivatives of Higher Order 235
　　　　Differentiation of Integrals 236
　　　　Exercises 239
　　　　Chapter 10 Integration of Differential Forms 245
　　　　Integration 245
　　　　Primitive Mappings 248
　　　　Partitions of Unity 251
　　　　Change of Variables 252
　　　　Differential Forms 253
　　　　Simplexes and Chains 266
　　　　Stokes’ Theorem 273
　　　　Closed Forms and Exact Forms 275
　　　　Vector Analysis 280
　　　　Exercises 288
　　　　Chapter 11 The Lebesgue Theory 300
　　　　Set Functions 300
　　　　Construction of the Lebesgue Measure 302
　　　　Measure Spaces 310
　　　　Measurable Functions 310
　　　　Simple Functions 313
　　　　Integration 314
　　　　Comparison with the Riemann Integral 322
　　　　Integration of Complex Functions 325
　　　　Functions of Class [WTHT]L[WT]\+2 325
　　　　Exercises 332
　　　　Bibliography 335
　　　　List of Special Symbols 337
　　　　Index 339