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General |
MATH 631 , Spring 2019: Notes and Summary
| Lecture | Date | What was covered | Notes | Textbook Section |
| 1 | Jan.29 | Metric Space (I) | 9.1, 9.2, 9.3 | |
| 2 | Jan.31 |
Metric Spaces(II) | Get HW1 | 9.4, 9.5 |
| 3 | Feb. 5 | Separability, Equicontinuity | 9.6, 10.1 | |
| 4 | Feb. 7 | Arzela-Ascoli Theorem | Due HW1; Get HW2 | 10.1 |
| 5 | Feb. 12 | Banach Contraction Principle | 10.3 | |
| 6 | Feb.14 | Baire Category Theorem | Due HW2; Get HW3 | 10.2 |
| 7 | Feb.19 | Topological Spaces (I) | 11.1 | |
| 8 | Feb.21 | Topological Spaces (II) | 11.2 | |
| 9 | Feb.26 | Topological Spaces (III) | Due HW3; Get HW4 | 11.3,11.4 |
| 10 | Feb.28 | Compactness | 11.5 | |
| 11 | Mar. 5 | Metrizability: Urysohn Theorem | class in MATH 1308 | 12.1 |
| 12 | Mar. 7 | Embedding: Urysohn Theorem. Product Toplogy; weak topology. | Due HW4; Get HW5 | 12.1, 11.4, 12.2 |
| 13 | Mar. 12 | Density: Stone-Weierstrass Theorem | 12.3 | |
| 14 | Mar.14 | Density: Stone-Weierstrass Theorem | Due HW5; Get HW6 | 12.3 |
| - | Mar.17-23 | SPRING BREAK | NO CLASSES | |
| 15 | Mar.26 | Compactness: Tychonoff Theorem. Product Topology | 12.2 | |
| 16 | Mar.28 | Tychonoff Theorem | Due HW6 | |
| 17 | Apr. 2 | Mid-Term Exam | Mid-Term Exam | |
| 18 | Apr. 4 | Norms, scalar products, Banach spaces, Hilbert spaces. | ||
| 19 | Apr. 9 | Duality Theory; | Get HW7 | 14.1 |
| 20 | Apr.11 | Closest points to closed convex sets in Hilbert spaces. | 16.1 | |
| 21 | Apr.16 | Riesz representation theorem for Hilbert spaces. The space C(X) for compact Hausdorff spaces X.Partitions of Unity. | Due HW7 | 21.1 |
| 22 | Apr.18 | Decomposition of bounded linear functionals. | Get HW8 | 21.5 |
| 23 | Apr.23 | Measures. Borel sigma-algebra B(X) | 17.1 | |
| 24 | Apr.25 | Baire sigma-algebra. Abstract measure spaces and measurability. | Due HW8. Get HW9 | 21.6, 17.1, 18.1 |
| 25 | Apr.30 | Abstract integral on measure spaces. | 18.2, 18.3 | |
| 26 | May 2 | Borel measure associated to a positive functional on C(X) | Due HW9. Get HW10 | 21.3 |
| 27 | May 7 | Borel Measure Construction | 17.3,17.4,17.5,21.3 | |
| 28 | May 9 | Riesz-Markov Theorem. Riesz-Fisher Theorem (completness of L1) | 21.5, 19.1 | |
| 29 | May 14 | Comparison of Measures: Radon-Nikodim Derivative. Absolute Continuity of Abstract Measures | Due HW10. LAST CLASS | 18.4 |
| - | Thur., May 16 | REVIEW SESSION | 10:30am-12:30pm | CSIC 4122 |
| - | Tue., May 21 | FINAL EXAM | CSIC 4122 | 1:30pm-3:30pm |
