| Spring 2020 MAT 533: Analysis II | ||
| Schedule | MW 10:00-11:20 am Math 4-130 | |
| Instructor | Robert Hough | |
| Office hours | F 9am-11pm in Math Tower 4-118, F 3-4pm in MLC. | |
| Description | Representations and decomposition theorems in measure theory; Fubini's theorem; L-p spaces; Fourier series; Laplace, heat and wave equations; open mapping and uniform boundedness theorems for Banach spaces; differentation of the integral; change of variable of integration. | |
| Textbook | Gerald B. Folland. Real Analysis. John Wiley and Sons (1999). | |
| Homework | Weekly problem sets will be assigned, and collected in class on Monday. | |
| Grading | The course grade is based upon the written homework (30%), a midterm (30%) and final exam (40%). | |
Syllabus/schedule (subject to change)
| Mon 1/27 | 1. | Review of point-set topology | Folland 4.1-4.2, 4.4-4.5 |
| Wed 1/29 | 2. | Compact and locally compact spaces | HW due 2/3: Homework 1, Solutions. |
| Mon 2/3 | 3. | Ascoli-Arzela Theorem, Stone-Weierstrass Theorem | Folland 4.6-4.7, 5.1-5.2 |
| Wed 2/5 | 4. | Normed vector spaces, linear functionals | HW due 2/10: Homework 2, Solutions. |
| Mon 2/10 | 5. | Baire Category Theorem | Folland 5.3-5.4 |
| Wed 2/12 | 6. | Topological vector spaces | HW due 2/17: Homework 3, Source file, Solutions. |
| Mon 2/17 | 7. | Hilbert space | Folland 5.5, 7.1 |
| Wed 2/19 | 8. | Positive linear functions | HW due 2/24: Homework 4, Source file, Solutions. |
| Mon 2/24 | 9. | Lusin's Theorem | Folland 7.2-7.3 |
| Wed 2/26 | 10. | The Riesz Representation Theorem | HW due 3/2: Homework 5, Source file, Solutions |
| Mon 3/2 | 11. | The Fubini-Tonelli Theorem | Folland 7.4, 8.1 |
| Wed 3/4 | 12. | Fourier preliminaries | HW due 3/9: Homework 6, Source file, Solutions |
| Mon 3/9 | 13. | Convolutions | Folland 8.2 |
| Wed 3/11 | Midterm I | Midterm solutions | |
| Mon 3/16 | No class - Spring Break | ||
| Wed 3/18 | No class - Spring Break | ||
| Mon 3/23 | No class - Spring Break | ||
| Wed 3/25 | No class - Spring Break | ||
| Mon 3/30 | 14. | The Fourier transform | Folland 8.3-8.4 |
| Wed 4/1 | 15. | Fourier inversion on the torus | HW due 4/6: Homework 7, Source file, Solutions |
| Mon 4/6 | 16. | Distributions | Folland 8.5-8.6 |
| Wed 4/8 | 17. | Sobolev spaces | HW due 4/13: Homework 8, Source file, Solutions |
| Mon 4/13 | 18. | Applications to PDE | Folland 8.7-8.8 |
| Wed 4/15 | 19. | Fourier analysis of measures | HW due 4/20: Homework 9, Source file, Solutions |
| Mon 4/20 | 20. | Basic concepts of probability | Folland 9.1-9.2 |
| Wed 4/22 | 21. | The law of large numbers | HW due 4/27: Homework 10, Source file, Solutions |
| Mon 4/27 | 22. | The central limit theorem | Folland 9.3-9.4 |
| Wed 4/29 | 23. | Construction of sample spaces | HW due 5/4: Homework 11, Source file, Solutions |
| Mon 5/4 | 24. | Brownian motion | Folland 9.5, 10.1 |
| Wed 5/6 | 25. | Haar measure | HW due 5/11: Homework 12, Source file, Solutions |
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