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General |
MATH 632 , Fall 2024: Notes and Summary
| Lecture | Date | What was covered | Notes | Textbook Section (R=Rudin; S I.=Simon vol.1; S IV. = Simon vol.4; RS=Reed and Simon - vol.1) |
| 1 | Aug.27 | Metric Spaces | CourseOverview , L1, get assignment 1 | (RS) I.2 |
| 2 | Aug.29 | Normed Vector Spaces, Hilbert Spaces | L2 | (RS)I.2, II.1 |
| 3 | Sep. 3 | Projection to closed convex subsets of Hilbert spaces; Normed vector space of bounded opeartors between NVS | L3, assignment 1 due, get assignment 2 | |
| 4 | Sep. 5 | Riesz Representation Theorem for Hilbert Spaces | L4, homework 1 due | (RS) II.2 |
| 5 | Sept.10 | Duality, ONB | L5, assignment 2 due | (RS) II.2, II.3 |
| 6 | Sept.12 | Convexity: Hahn-Banach (real case) | L6 | (S I.) 5.5 |
| 7 | Sept.17 | Complex Hahn-Banach; Double Dual | homework 2 due, L7 | (S I.) 5.5 |
| 8 | Sept.19 | Topological spaces. Dual topologies | L8 | (R I.) 5.7, (RS) VI.1 |
| 9 | Sept.24 | Baire Category Theorem | L9 | (RS) III.5 |
| 10 | Sept.26 | Banach-Steinhaus Thm, Open Mapping Theorem | L10 | (RS)III.5 |
| 11 | Oct. 1 | Inverse Mapping Theorem, Lower Bound Theorem, Closed Graph Theorem, | L11 HW4 due | (RS) III.4 |
| 12 | Oct. 3 | Spectrum of operators. Compact operators. | L12 | (S I.)5.1, (S IV.)3.1, (RS)VI.3 |
| 13 | Oct. 8 | Compact Operators (2) | L13 | (R) 4.18, 4.19 |
| 14 | Oct.10 | Compact Operators (3). Weak compactness of closed unit ball in Hilbert spaces. | L14 | (RS)VI.5 |
| 15 | Oct.15 | Spectral Theorem for Compact Operators (1) | L15 | (RS) VI.5 |
| 16 | Oct.17 | SVD for compact operators | L16 | |
| 17 | Oct.22 | Schatten Classes of Compact Operators | L17 HW5 due | (RS) VI.5 |
| 18 | Oct.24 | Mid-Term Exam | - | |
| 19 | Oct.29 | Banach Algebras | L18 | (S IV)6.1 |
| 20 | Oct.31 | Gelfand's Formula | L19 | (S IV)6.2 |
| 21 | Nov. 5 | Holomorphic Calculus (1) | L20, HW6 due | (S IV)2.3 |
| 22 | Nov. 7 | Holomorphic Calculus (2) | L21 | (S IV)2.3 |
| 23 | Nov. 12 | Holomorphic Calculus (3) | L22 | (S IV) 2.3 |
| 24 | Nov.14 | Operators on Hilbert Spaces | L24, HW7 due | (S IV) 2.4 |
| 25 | Nov.19 | Square Root Lemma | L25, | (S IV) 2.4 |
| 26 | Nov.21 | Polar Decomposition | L26 | (S IV) 2.4 |
| 27 | Nov.26 | Spectral Calculus: Resolution of Identity. Continuous Functional Calculus | L27, HW8 due | |
| - | Nov.28 | THANKSGIVING BREAK | NO CLASS | |
| 28 | Dec. 3 | Spectral Calculus: Borel Functional Calculus (1) | L28 | (S IV) 5 |
| 29 | Dec. 5 | Spectral Calculus: Borel Calculus (2) | L29 , Last Day of Classes | (S IV) 5 |
| - | Monday, Dec.16 | FINAL EXAM | 4-6pm | Classroom: B0431 |
