Spring 2022 MAT 533: Analysis II | ||
Schedule | MW 2:40-4:00 pm Physics P112 | |
Instructor | Robert Hough | |
Office hours | Th 9am-11pm in Math Tower 4-118, W 7-8pm 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. Submit the homework each week by Midnight on Monday to Matthew Dannenberg, matthew.dannenberg@stonybrook.edu. | |
Grading | The course grade is based upon the written homework (30%), a midterm (30%) and final exam (40%). |
Syllabus/schedule (subject to change)
Mon 1/24 | 1. | Review of point-set topology | Folland 4.1-4.2 HW due 1/31: Homework 1. |
Wed 1/26 | 2. | Compact and locally compact spaces | Folland 4.4-4.5 |
Mon 1/31 | 3. | Ascoli-Arzela Theorem, Stone-Weierstrass Theorem | Folland 4.6-4.7 HW due 2/7: Homework 2. |
Wed 2/2 | 4. | Normed vector spaces, linear functionals | Folland 5.1-5.2 |
Mon 2/7 | 5. | Baire Category Theorem | Folland 5.3 HW due 2/14: Homework 3 |
Wed 2/9 | 6. | Topological vector spaces | Folland 5.4 |
Mon 2/14 | 7. | Hilbert space | Folland 5.5 HW due 2/21: Homework 4 |
Wed 2/16 | 8. | Positive linear functions | Folland 7.1 |
Mon 2/21 | 9. | Lusin's Theorem | Folland 7.2 HW due 2/28: Homework 5 |
Wed 2/23 | 10. | The Riesz Representation Theorem | Folland 7.3 |
Mon 2/28 | 11. | The Fubini-Tonelli Theorem | Folland 7.4 HW due 3/7: Homework 6 |
Wed 3/2 | 12. | Fourier preliminaries | Folland 8.1 |
Mon 3/7 | 13. | Convolutions | Folland 8.2 HW due 3/21: Homework 7 |
Wed 3/9 | 14. | The Fourier Transform | Folland 8.3 |
Mon 3/14 | No class - Spring Break | ||
Wed 3/16 | No class - Spring Break | ||
Mon 3/21 | Midterm | Practice Midterm | |
Wed 3/23 | 15. | Fourier inversion on the torus | Folland 8.4 |
Mon 3/28 | 16. | Distributions | Folland 8.5 HW due 4/4: Homework 8 |
Wed 3/30 | 17. | Sobolev spaces | Folland 8.6 |
Mon 4/4 | 18. | Applications to PDE | Folland 8.7 |
Wed 4/6 | 19. | Fourier analysis of measures | Folland 8.8 |
Mon 4/11 | 20. | Basic concepts of probability | Folland 9.1 HW due 4/18: Homework 9 |
Wed 4/13 | 21. | The law of large numbers | Folland 9.2 |
Mon 4/18 | 22. | The central limit theorem | Folland 9.3 HW due 4/25: Homework 10 |
Wed 4/20 | 23. | Construction of sample spaces | Folland 9.4 |
Mon 4/25 | 24. | Brownian motion | Folland 9.5 HW due 5/2: Homework 11 |
Wed 4/27 | 25. | Haar measure | Folland 10.1 |
Mon 5/2 | 26. | Hausdorff Measure | Folland 10.2 |
Wed 5/4 | 27. | Integration on manifolds | Folland 10.3 |
Disability Support Services: If you have a physical, psychological, medical, or learning disability that may affect your course work, please contact Disability Support Services (DSS) office: ECC (Educational Communications Center) Building, room 128, telephone (631) 632-6748/TDD. DSS will determine with you what accommodations are necessary and appropriate. Arrangements should be made early in the semester (before the first exam) so that your needs can be accommodated. All information and documentation of disability is confidential. Students requiring emergency evacuation are encouraged to discuss their needs with their professors and DSS. For procedures and information, go to the following web site http://www.ehs.stonybrook.edu and search Fire safety and Evacuation and Disabilities.
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