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Spaces is a modern introduction to real analysis at the advanced undergraduate level. It is forward-looking in the sense that it first and foremost aims to provide students with the concepts and techniques they need in order to follow more advanced courses in mathematical analysis and neighboring fields. The only prerequisites are a solid understanding of calculus and linear algebra. Two introductory chapters will help students with the transition from computation-based calculus to theory-based analysis.
The main topics covered are metric spaces, spaces of continuous functions, normed spaces, differentiation in normed spaces, measure and integration theory, and Fourier series. Although some of the topics are more advanced than what is usually found in books of this level, care is taken to present the material in a way that is suitable for the intended audience: concepts are carefully introduced and motivated, and proofs are presented in full detail. Applications to differential equations and Fourier analysis are used to illustrate the power of the theory, and exercises of all levels from routine to real challenges help students develop their skills and understanding. The text has been tested in classes at the University of Oslo over a number of years.
Tom L. Lindstrøm, University of Oslo, Oslo, Norway
• Preface 10 • Introduction –Mainly to the Students 14 • Chapter 1. Preliminaries: Proofs, Sets, and Functions 18 1.1. Proofs 18 1.2. Sets and Boolean operations 21 1.3. Families of sets 24 1.4. Functions 26 1.5. Relations and partitions 30 1.6. Countability 33 Notes and references for Chapter 1 35 • Chapter 2. The Foundation of Calculus 36 2.1. Epsilon-delta and all that 37 2.2. Completeness 42 2.3. Four important theorems 50 Notes and references for Chapter 2 55 • Chapter 3. Metric Spaces 56 3.1. Definitions and examples 56 3.2. Convergence and continuity 61 3.3. Open and closed sets 65 3.4. Complete spaces 72 3.5. Compact sets 76 3.6. An alternative description of compactness 81 3.7. The completion of a metric space 84 Notes and references for Chapter 3 89 • Chapter 4. Spaces of Continuous Functions 92 4.1. Modes of continuity 92 4.2. Modes of convergence 94 4.3. Integrating and differentiating sequences 99 4.4. Applications to power series 105 4.5. Spaces of bounded functions 112 4.6. Spaces of bounded, continuous functions 114 4.7. Applications to differential equations 116 4.8. Compact sets of continuous functions 120 4.9. Differential equations revisited 125 4.10. Polynomials are dense in the continuous function 129 4.11. The Stone-Weierstrass Theorem 136 Notes and references for Chapter 4 144 • Chapter 5. Normed Spaces and Linear Operators 146 5.1. Normed spaces 146 5.2. Infinite sums and bases 153 5.3. Inner product spaces 155 5.4. Linear operators 163 5.5. Inverse operators and Neumann series 168 5.6. Baire’s Category Theorem 174 5.7. A group of famous theorems 180 Notes and references for Chapter 5 184 • Chapter 6. Differential Calculus in Normed Spaces 186 6.1. The derivative 187 6.2. Finding derivatives 195 6.3. The Mean Value Theorem 200 6.4. The Riemann Integral 203 6.5. Taylor’s Formula 207 6.6. Partial derivatives 214 6.7. The Inverse Function Theorem 219 6.8. The Implicit Function Theorem 225 6.9. Differential equations yet again 229 6.10. Multilinear maps 239 6.11. Higher order derivatives 243 Notes and references for Chapter 6 251 • Chapter 7. Measure and Integration 252 7.1. Measure spaces 253 7.2. Complete measures 261 7.3. Measurable functions 265 7.4. Integration of simple functions 270 7.5. Integrals of nonnegative functions 275 7.6. Integrable functions 284 7.7. Spaces of integrable functions 289 7.8. Ways to converge 298 7.9. Integration of complex functions 301 Notes and references for Chapter 7 303 • Chapter 8. Constructing Measures 304 8.1. Outer measure 305 8.2. Measurable sets 307 8.3. Carathéodory’s Theorem 310 8.4. Lebesgue measure on the real line 317 8.5. Approximation results 320 8.6. The coin tossing measure 324 8.7. Product measures 326 8.8. Fubini’s Theorem 329 Notes and references for Chapter 8 337 • Chapter 9. Fourier Series 338 9.1. Fourier coefficients and Fourier series 340 9.2. Convergence in mean square 346 9.3. The Dirichlet kernel 349 9.4. The Fejér kernel 354 9.5. The Riemann-Lebesgue Lemma 360 9.6. Dini’s Test 363 9.7. Pointwise divergence of Fourier series 367 9.8. Termwise operations 369 Notes and references for Chapter 9 372 • Bibliography 374 • Index 376