| Fall 2021 MAT 324: Real analysis | ||
| Schedule | TTh 9:45-11:05am Library E4330 | |
| Instructor | Robert Hough | |
| Office hours | Tu 7-8 in MLC, F 9am-11pm in Math Tower 4-118. | |
| Description | Introduction to Lebesgue measure and integration. Aspects of Fourier series, function spaces, Hilbert spaces, Banach spaces. | |
| Textbook | Sheldon Axler. Measure, integration and real analysis. Springer (2020). | |
| Grading | The course grade is based upon the written homework. | |
Syllabus/schedule (subject to change)
| Tues 8/24 | 1. | Course overview, outer measure, measure space | Axler 2A, 2B | Homework 1: p.23 #9, 10, p.38 #14, 25, p.45 #1, 2, p.60 #5, 13, Solutions |
| Thurs 8/26 | 2. | Measures, Lebesgue Measure | Axler 2C, 2D | |
| Tues 8/31 | 3. | Egoroff and Lusin's Theorems, Monotone Convergence | Axler 2E, 3A | Homework 2: p.71 #7, 8, p.84 #3, 4, 8, p.99 #4, 5, 7, Solutions |
| Thurs 9/2 | 4. | Bounded convergence theorem, dominated convergence theorem, properties of the integral | Axler 3B | |
| Tues 9/7 | 5. | Markov's inequality, Vitali covering lemma, Hardy-Littlewood Maximal Function | Axler 4A | Homework 3: p.106 #4, 8, 11, 13, p.115 #2, 3, 9, Solutions |
| Thurs 9/9 | 6. | Lebesgue Diff. Theorem | Axler 4B | |
| Tues 9/14 | 7. | Products of measure spaces | Axler 5A | Homework 4: p.128 #1, 4, 8, 9, p.135 #1, 4 Solutions |
| Thurs 9/16 | 8. | Tonelli and Fubini Theorems | Axler 5B | |
| Tues 9/21 | 9. | Integration in n-space | Axler 5C | Homework 5: p.144 #4, 5, p.153 #4, 5, 6, p.162 #6, 7 Solutions |
| Thurs 9/23 | 10. | Metric spaces, vector spaces | Axler 6A, 6B | |
| Tues 9/28 | 11. | Normed vector spaces | Axler 6C | Homework 6: p.170 #7, 8, 9, 15, p.181 #2, 3, 18, 20 Solutions |
| Thurs 9/30 | 12. | Linear functionals, Hahn-Banach Theorem | Axler 6D | |
| Tues 10/5 | 13. | Baire Category, Uniform Boundedness Principle | Axler 6E | Homework 7: p.190 #5, 8, 9, 16, p.199 #5, 7, 11, 17 Solutions |
| Thurs 10/7 | 14. | Hölder and Minkowski Inequalities | Axler 7A | |
| Tues 10/12 | No class - Fall Break | |||
| Thurs 10/14 | 15. | Review lecture | ||
| Tues 10/19 | 16. | L^p and dual spaces | Axler 7B | Homework 8: p.208 #6, 11, 15, p.221 #1, 5, 7, 15, p.234 #11, 13, 22 Solutions |
| Thurs 10/21 | 17. | Hilbert space | Axler 8A,8B | |
| Tues 10/26 | 18. | Riesz Representation Theorem | Axler 8C | Homework 9: p.251 #2, 6, 12, 24, p.265 #3, 4, 11 Solutions |
| Thurs 10/28 | 19. | Real and complex measures | Axler 9A | |
| Tues 11/2 | 20. | Hahn decomposition, Jordan Decomposition, Lebesgue Decomposition, Radon-Nikodym Theorem | Axler 9B | Homework 10: p.278 #1, 2, 11, 13, p.292 #3, 9, 10 Solutions |
| Thurs 11/4 | 21. | Adjoint and Inverse Operators | Axler 10A | |
| Tues 11/9 | 22. | Spectrum, spectral mapping theorem, isometries | Axler 10B | Homework 11: p.309 #2, 7, 10, 23, p.323 #1, 6, 7, 10 Solutions |
| Thurs 11/11 | 23. | Compact operators, Fredholm alternative | Axler 10C | |
| Tues 11/16 | 24. | Spectral theorem, singular value decomposition | Axler 10D | Homework 12: p.336 #4, 7, 12, 17, p.352 #4, 6, 10, 12 Solutions |
| Thurs 11/18 | 25. | Fourier series and Poisson integral | Axler 11A | |
| Tues 11/23 | 26. | Fourier series in L^p | Axler 11B | Homework 13: p.361 #9, 11, 15, 18, p.377 #5, 6, 8, 9 Solutions |
| Thurs 11/25 | No class - Thanksgiving | |||
| Tues 11/30 | 27. | The Fourier transform in L^1(R) | Axler 11C | Homework 14: p.398 #4, 6, 9, 17 |
| Thurs 12/2 | 28. | Probability Measures | Axler 12 |
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