| Preface | p. xi |
| Metric Spaces | |
| Calculus Review | p. 3 |
| The Real Numbers | p. 3 |
| Limits and Continuity | p. 14 |
| Notes and Remarks | p. 17 |
| Countable and Uncountable Sets | p. 18 |
| Equivalence and Cardinality | p. 18 |
| The Cantor Set | p. 25 |
| Monotone Functions | p. 31 |
| Notes and Remarks | p. 34 |
| Metrics and Norms | p. 36 |
| Metric Spaces | p. 37 |
| Normed Vector Spaces | p. 39 |
| More Inequalities | p. 43 |
| Limits in Metric Spaces | p. 45 |
| Notes and Remarks | p. 49 |
| Open Sets and Closed Sets | p. 51 |
| Open Sets | p. 51 |
| Closed Sets | p. 53 |
| The Relative Metric | p. 60 |
| Notes and Remarks | p. 62 |
| Continuity | p. 63 |
| Continuous Functions | p. 63 |
| Homeomorphisms | p. 69 |
| The Space of Continuous Functions | p. 73 |
| Notes and Remarks | p. 76 |
| Connectedness | p. 78 |
| Connected Sets | p. 78 |
| Notes and Remarks | p. 87 |
| Completeness | p. 89 |
| Totally Bounded Sets | p. 89 |
| Complete Metric Spaces | p. 92 |
| Fixed Points | p. 97 |
| Completions | p. 102 |
| Notes and Remarks | p. 106 |
| Compactness | p. 108 |
| Compact Metric Spaces | p. 108 |
| Uniform Continuity | p. 114 |
| Equivalent Metrics | p. 120 |
| Notes and Remarks | p. 126 |
| Category | p. 128 |
| Discontinuous Functions | p. 128 |
| The Baire Category Theorem | p. 131 |
| Notes and Remarks | p. 136 |
| Function Spaces | |
| Sequences of Functions | p. 139 |
| Historical Background | p. 139 |
| Pointwise and Uniform Convergence | p. 143 |
| Interchanging Limits | p. 150 |
| The Space of Bounded Functions | p. 153 |
| Notes and Remarks | p. 160 |
| The Space of Continuous Functions | p. 162 |
| The Weierstrass Theorem | p. 162 |
| Trigonometric Polynomials | p. 170 |
| Infinitely Differentiable Functions | p. 176 |
| Equicontinuity | p. 178 |
| Continuity and Category | p. 183 |
| Notes and Remarks | p. 185 |
| The Stone-Weierstrass Theorem | p. 188 |
| Algebras and Lattices | p. 188 |
| The Stone-Weierstrass Theorem | p. 194 |
| Notes and Remarks | p. 201 |
| Functions of Bounded Variation | p. 202 |
| Functions of Bounded Variation | p. 202 |
| Helly's First Theorem | p. 210 |
| Notes and Remarks | p. 212 |
| The Riemann-Stieltjes Integral | p. 214 |
| Weights and Measures | p. 214 |
| The Riemann-Stieltjes Integral | p. 215 |
| The Space of Integrable Functions | p. 221 |
| Integrators of Bounded Variation | p. 225 |
| The Riemann Integral | p. 232 |
| The Riesz Representation Theorem | p. 234 |
| Other Definitions, Other Properties | p. 239 |
| Notes and Remarks | p. 242 |
| Fourier Series | p. 244 |
| Preliminaries | p. 244 |
| Dirichlet's Formula | p. 250 |
| Fejer's Theorem | p. 254 |
| Complex Fourier Series | p. 257 |
| Notes and Remarks | p. 258 |
| Lebesgue Measure and Integration | |
| Lebesgue Measure | p. 263 |
| The Problem of Measure | p. 263 |
| Lebesgue Outer Measure | p. 268 |
| Riemann Integrability | p. 274 |
| Measurable Sets | p. 277 |
| The Structure of Measurable Sets | p. 283 |
| A Nonmeasurable Set | p. 289 |
| Other Definitions | p. 292 |
| Notes and Remarks | p. 293 |
| Measurable Functions | p. 296 |
| Measurable Functions | p. 296 |
| Extended Real-Valued Functions | p. 302 |
| Sequences of Measurable Functions | p. 304 |
| Approximation of Measurable Functions | p. 306 |
| Notes and Remarks | p. 310 |
| The Lebesgue Integral | p. 312 |
| Simple Functions | p. 312 |
| Nonnegative Functions | p. 314 |
| The General Case | p. 322 |
| Lebesgue's Dominated Convergence Theorem | p. 328 |
| Approximation of Integrable Functions | p. 333 |
| Notes and Remarks | p. 335 |
| Additional Topics | p. 337 |
| Convergence in Measure | p. 337 |
| The L[subscript p] Spaces | p. 342 |
| Approximation of L[subscript p] Functions | p. 350 |
| More on Fourier Series | p. 352 |
| Notes and Remarks | p. 356 |
| Differentiation | p. 359 |
| Lebesgue's Differentiation Theorem | p. 359 |
| Absolute Continuity | p. 370 |
| Notes and Remarks | p. 377 |
| References | p. 379 |
| Symbol Index | p. 395 |
| Topic Index | p. 397 |
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