Real analysis

PART ONE. METRIC SPACES......Page 12
The Real Numbers......Page 14
Limits and Continuity......Page 25
Notes and Remarks......Page 28
Equivalence and Cardinality......Page 29
The Cantor Set......Page 36
Monotone Functions......Page 42
Notes and Remarks......Page 45
3 Metrics and Norms......Page 47
Metric Spaces......Page 48
Normed Vector Spaces......Page 50
More Inequalities......Page 54
Limits in Metric Spaces......Page 56
Notes and Remarks......Page 60
Open Sets......Page 62
Closed Sets......Page 64
The Relative Metric......Page 71
Notes and Remarks......Page 73
Continuous Functions......Page 74
Homeomorphisms......Page 80
The Space of Continuous Functions......Page 84
Notes and Remarks......Page 87
Connected Sets......Page 89
Notes and Remarks......Page 98
Totally Bounded Sets......Page 100
Complete Metric Spaces......Page 103
Fixed Points......Page 108
Completions......Page 113
Notes and Remarks......Page 117
8 Compactness......Page 119
Helly's First Theorem 2]0......Page
Uniform Continuity......Page 125
Equivalent Metrics......Page 131
Notes and Remarks......Page 137
Discontinuous Functions......Page 139
The Baire Category Theorem......Page 142
Notes and Remarks......Page 147
PART TWO. FUNCTION SPACES......Page 148
Historical Background......Page 150
Pointwise and Uniform Convergence......Page 154
Interchanging Limits......Page 161
The Space of Bounded Functions......Page 164
Notes and Remarks......Page 171
The Weierstrass Theorem......Page 173
Trigonometric Polynomials......Page 181
Infinitely Differentiable Functions......Page 187
Equicontinuity......Page 189
Continuity and Category......Page 194
Notes and Remarks......Page 196
Algebras and Lattices......Page 199
The Stone-Weierstrass Theorem......Page 205
Notes and Remarks......Page 212
Functions of Bounded Variation......Page 213
Notes and Remarks......Page 223
Weights and Measures......Page 225
The Riemann-Stieltjes Integral......Page 226
The Space of Integrable Functions......Page 232
Integrators of Bounded Variation......Page 236
The Riemann Integral......Page 243
The Riesz Representation Theorem......Page 245
Other Definitions, Other Properties......Page 250
Notes and Remarks......Page 253
Preliminaries......Page 255
Dirichlet's Fonnula......Page 261
Fejer's Theorem......Page 265
Complex Fourier Series......Page 268
Notes and Remarks......Page 269
PART THREE. LEBESGUE MEASURE AND INTEGRATION......Page 272
The Problem of Measure......Page 274
Lebesgue Outer Measure......Page 279
Riemann Integrability......Page 285
Measurable Sets......Page 288
The Structure of Measurable Sets......Page 294
A Nonmeasurable Set......Page 300
Other Definitions......Page 303
Notes and Remarks......Page 304
Measurable Functions......Page 307
Extended Real-Valued Functions......Page 313
Sequences of Measurable Functions......Page 315
Approximation of Measurable Functions......Page 317
Notes and Remarks......Page 321
Simple Functions......Page 323
Nonnegative Functions......Page 325
The General Case......Page 333
Lebesgue's Dominated Convergence Theorem......Page 339
Approximation of Integrable Functions......Page 344
Notes and Remarks......Page 346
Convergence in Measure......Page 348
The Lp Spaces......Page 353
Approximation of Lp Functions......Page 361
More on Fourier Series......Page 363
Notes and Remarks......Page 367
Lebesgue's Differentiation Theorem......Page 370
Absolute Continuity......Page 381
Notes and Remarks......Page 388
References......Page 390
Symbol Index......Page 406
Topic Index......Page 408