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General |
MATH 630 , Fall 2017: Notes and Summary
| Lecture | Date | What was covered | Notes | Textbook Section |
| 1 | Aug. 29 | Numbers (N,Z,Q,R,C). Relations (equivalence, order) | get assignment 1 | 1.1, 1.3, 1.4 |
| 2 | Aug. 31 | Metric spaces. Completion of rationals. | - | 1.2, 1.5 |
| 3 | Sep. 5 | (Axiom of Choice and Zorn's Lemma. Countability.) Metric Spaces. Topology. Topological Spaces | return assignment 1; get assignment 2 | 1.2 (1.5, 1.6) 2.1 |
| 4 | Sep. 7 | Completion of Metric Spaces | ||
| 5 | Sept.12 | Norms, Banach Spaces, Separability, Scalar products, Hilbert Spaces (I) | return assignment 2; get assignment 3 | 3.1 |
| 6 | Sept.14 | Hilbert Spaces (II): orthonormality, Pythagora, Bessel, Cauchy-Schwarz, non-separability of l^inf | 3.2 | |
| 7 | Sept.19 | completion of l^p | return assignment 3; get assignment 4 | |
| 8 | Sept.21 | Duality in Hilbert Spaces. Riesz Representation theorem | 3.2, 3.3 | |
| 9 | Sept.26 | Riemeann-Stieltjes Integral (1) | return assignment 4; get assignment 5 | 4.1 |
| 10 | Sept.28 | Functions in BV space. Jordan decomposition. Pure point part of a BV function | 4.15 | |
| 11 | Oct. 3 | Riemann-Stieltjes integral (2) | return assignment 5; get assignment 6 | 4.1 |
| 12 | Oct. 5 | Cantor sets. Cantor function. | 4.2 | |
| 13 | Oct.10 | Cantor Function (2) | get assignment 7 | 4.2 , MATH 3206 |
| 14 | Oct.12 | Measurable spaces and functions. Borel and Baire Sigma algebras. | return assignment 6 | 4.3 (and Chapter 2 for Gdelta, Fsigma sets), MATH 3206 |
| 15 | Oct.17 | L^1 as completion of C[a,b] | return assignment 7 | 4.4 |
| - | Oct.18 | REVIEW SESSION | 3:15pm-4:50pm | CSIC 4122 |
| 16 | Oct.19 | MID TERM EXAM | ||
| 17 | Oct.24 | Lipschitz maps acting on L^1. Monotone Convergence Theorem (1). Partition of Unity | get assignment 8 | 4.4 |
| 18 | Oct.26 | Measure of Open Sets. Additivity and Subadditivity. | 4.4 | |
| 19 | Oct.31 | Riesz-Fischer Theorem. Luzin Theorem. | return assignment 8; get assignment 9 | 4.4 |
| 20 | Nov. 2 | Luzin and Egorov theorems for L1 functions. Characterizations of abstract L1 space | 4.4 | |
| 21 | Nov. 7 | Monotone Convergence Theorem (2); Measure of Borel sets. | return assignment 9; pick up assignment 10 | 4.4 |
| 22 | Nov. 9 | Regularity of Borel measures. | 4.4 and 4.5 (Theorem 4.5.6) | |
| 23 | Nov.14 | Lebesgue and Lebesgue-Stieljes integrals of L1 functions. Lp and Linf spaces. | return assignment 10; get assignment 11 | 4.4 |
| 24 | Nov. 16 | Convergence Theorems: Fatou Lemma, Lebegues Dominated Convergence | 4.6 | |
| 25 | Nov.21 | Brezis-Lieb Theorem, Scheffe Lemma; Egorov Theorem | return assignment 11; get assignment 12 | 4.6 |
| - | Nov.23 | THANKSGIVING BREAK | NO CLASS | |
| 26 | Nov.28 | Product Measure. Fubini Theorem | 4.11 | |
| 27 | Nov.30 | Absolute continuity. Mutually singular measures. | return assignment 12; get assignment 13 | 4.7 |
| 28 | Dec. 5 | VonNeumann theorem. Lebesgue decomposition theorem. Radon-Nikodym theorem. | 4.7 | |
| 29 | Dec. 7 | Absolute Continuity for measures and BV functions | return assignment 13 | LAST CLASS |
| - | Thurs., Dec.14 | REVIEW SESSION | 12:30pm-2:00pm | CSIC 4122 |
| - | Mon, Dec. 18 | FINAL EXAM | CSIC 4122 | 1:30pm-3:30pm |
