Math 528, Banach and Hilbert Spaces
Fall 2020
Topics, Exams and Homework Assignments
| Week | Dates | Topics | Sections Covered | Homework | Due |
|---|---|---|---|---|---|
| 1 | Aug 24—Aug 28 | Linear Spaces. Linear Maps. Algebra of linear maps. Index of a linear map. | Chapter 1, 2.1, 2.2 | Chapter 1: 1.7, 1.8, 2.1: 2.1.3, 2.2: 2.2.9 H1: H1 | Sep 4 |
| 2 | Aug 31—Sep 4 | Flag of nullspaces of powers of a linear operator. Nilpotent operators. Jordan canonical form. Index of a linear map. The Hahn-Banach Theorem. | 2.1, 2.2, 3.1 | Sep 11 | |
| 3 | Sep 7 | Labor Day - no class. | |||
| 3 | Sep 9—Sep 11 | The Hahn-Banach Theorem. The extension theorem. Geometric Hahn-Banach theorem. Extensions of the Hahn-Banach theorem. Applications of the Hahn-Banach theorem. Extension of positive linear functions. Banach limits. Finitely additive invariant set functions. | 3.1, 3.2, 3.3, 4.1, 4.2, 4.3 | 3.2: 3.2.2, 3.3: 3.3.3 H2: H2 | Sep 13 |
| 4 | Sep 14—Sep 18 | Normed linear spaces. Norms. Noncompactness of the unit ball. Isometries. | 5.1, 5.2, 5.3 | Sep 25 | |
| 5 | Sep 21—Sep 25 | Hilbert space. Scalar product. Closest point in a closed convex subset. Linear functionals. Linear span. | 6.1, 6.2, 6.3, 6.4 | 5.1: 5.1.3 H3: H3 | Sep 30 |
| 6 | Sep 28—Oct 2 | Applications of Hilbert space results. Radon-Nikodym theorem. | 7.1 | H4: H4 | Oct 9 |
| 7 | Oct 5—Oct 9 | Measure theory review. Radon-Nikodym theorem. | 7.1 | Oct 16 | |
| 8 | Oct 12—Oct 16 | Dirichlet's problem. | 7.2, 7.3 | Oct 23 | |
| 9 | Oct 21 | Midterm 1. | |||
| 9 | Oct 24—Oct 28 | Dirichlet's problem. | 7.2, 7.3 | ||
| 10 | Oct 26—Oct 30 | Duals of normed linear spaces. Bounded linear functionals. Extension of bounded linear functionals. Reflexive spaces. Support function of a set. | 8.1, 8.2, 8.3, 8.4 | H5: H5 | Nov 6 |
| 11 | Nov 2—Nov 6 | Applications of duality. Completeness of weighted powers. The Müntz approximation theorem. | 9.1, 9.2 | Nov 14 | |
| 12 | Nov 11 | Veteran's Day - no class. | |||
| 12 | Nov 9—Nov 13 | The weak and weak* topologies. Week convergence. Week sequential compactness. Week-*-convergence. | 10.1, 10.2, 10.3 | H6: H6 | |
| 13 | Nov 16—Nov 20 | The weak and weak* topologies. Weak convergence. Weak sequential compactness. Weak-*-convergence. | 10.1, 10.2, 10.3 | ||
| 14 | Nov 23—Nov 27 | The weak and weak* topologies. Week convergence. Week sequential compactness. Week-*-convergence. | 10.1, 10.2, 10.3 | ||
| 15 | Nov 26—Nov 29 | Thanksgiving recess. | |||
| 16 | Nov 30—Dec 4 | The weak and weak* topologies. Week convergence. Week sequential compactness. Week-*-convergence. | 10.1, 10.2, 10.3 | H7 (Midterm 2): H7 | |
| 16 | Dec 2 | Midterm 2. | |||
| 17 | Dec 7—Dec 9 | Review. Problem solving. | |||
| 17 | Dec 9 | Last Day of classes | |||
| 17 | Dec 10 | Reading Day - no classes or finals | |||
| Finals Week (Dec 11-17) | Dec 14 (Monday) | Final Exam, 1:00 pm - 3:00 pm (take-home, to be made available on Dec. 7) |