Spaces: an introduction to real analysis

Cover......Page 1
Title page......Page 4
Contents......Page 6
Preface......Page 10
Introduction –Mainly to the Students......Page 14
 1.1. Proofs......Page 18
 1.2. Sets and Boolean operations......Page 21
 1.3. Families of sets......Page 24
 1.4. Functions......Page 26
 1.5. Relations and partitions......Page 30
 1.6. Countability......Page 33
 Notes and references for Chapter 1......Page 35
Chapter 2. The Foundation of Calculus......Page 36
 2.1. Epsilon-delta and all that......Page 37
 2.2. Completeness......Page 42
 2.3. Four important theorems......Page 50
 Notes and references for Chapter 2......Page 55
 3.1. Definitions and examples......Page 56
 3.2. Convergence and continuity......Page 61
 3.3. Open and closed sets......Page 65
 3.4. Complete spaces......Page 72
 3.5. Compact sets......Page 76
 3.6. An alternative description of compactness......Page 81
 3.7. The completion of a metric space......Page 84
 Notes and references for Chapter 3......Page 89
 4.1. Modes of continuity......Page 92
 4.2. Modes of convergence......Page 94
 4.3. Integrating and differentiating sequences......Page 99
 4.4. Applications to power series......Page 105
 4.5. Spaces of bounded functions......Page 112
 4.6. Spaces of bounded, continuous functions......Page 114
 4.7. Applications to differential equations......Page 116
 4.8. Compact sets of continuous functions......Page 120
 4.9. Differential equations revisited......Page 125
 4.10. Polynomials are dense in the continuous function......Page 129
 4.11. The Stone-Weierstrass Theorem......Page 136
 Notes and references for Chapter 4......Page 144
 5.1. Normed spaces......Page 146
 5.2. Infinite sums and bases......Page 153
 5.3. Inner product spaces......Page 155
 5.4. Linear operators......Page 163
 5.5. Inverse operators and Neumann series......Page 168
 5.6. Baire’s Category Theorem......Page 174
 5.7. A group of famous theorems......Page 180
 Notes and references for Chapter 5......Page 184
Chapter 6. Differential Calculus in Normed Spaces......Page 186
 6.1. The derivative......Page 187
 6.2. Finding derivatives......Page 195
 6.3. The Mean Value Theorem......Page 200
 6.4. The Riemann Integral......Page 203
 6.5. Taylor’s Formula......Page 207
 6.6. Partial derivatives......Page 214
 6.7. The Inverse Function Theorem......Page 219
 6.8. The Implicit Function Theorem......Page 225
 6.9. Differential equations yet again......Page 229
 6.10. Multilinear maps......Page 239
 6.11. Higher order derivatives......Page 243
 Notes and references for Chapter 6......Page 251
Chapter 7. Measure and Integration......Page 252
 7.1. Measure spaces......Page 253
 7.2. Complete measures......Page 261
 7.3. Measurable functions......Page 265
 7.4. Integration of simple functions......Page 270
 7.5. Integrals of nonnegative functions......Page 275
 7.6. Integrable functions......Page 284
 7.7. Spaces of integrable functions......Page 289
 7.8. Ways to converge......Page 298
 7.9. Integration of complex functions......Page 301
 Notes and references for Chapter 7......Page 303
Chapter 8. Constructing Measures......Page 304
 8.1. Outer measure......Page 305
 8.2. Measurable sets......Page 307
 8.3. Carathéodory’s Theorem......Page 310
 8.4. Lebesgue measure on the real line......Page 317
 8.5. Approximation results......Page 320
 8.6. The coin tossing measure......Page 324
 8.7. Product measures......Page 326
 8.8. Fubini’s Theorem......Page 329
 Notes and references for Chapter 8......Page 337
Chapter 9. Fourier Series......Page 338
 9.1. Fourier coefficients and Fourier series......Page 340
 9.2. Convergence in mean square......Page 346
 9.3. The Dirichlet kernel......Page 349
 9.4. The Fejér kernel......Page 354
 9.5. The Riemann-Lebesgue Lemma......Page 360
 9.6. Dini’s Test......Page 363
 9.7. Pointwise divergence of Fourier series......Page 367
 9.8. Termwise operations......Page 369
 Notes and references for Chapter 9......Page 372
Bibliography......Page 374
Index......Page 376
Back Cover......Page 384