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General |
MATH 632 , Fall 2020: 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 | Sep. 1 | Metric Spaces | CourseOverview , L1 | (RS) I.2 |
| 2 | Sep. 3 | Normed Vector Spaces, Hilbert Spaces | L2, get homework 1 | (RS)I.2, II.1 |
| 3 | Sep. 8 | Projection to closed convex subsets of Hilbert spaces; Normed vector space of bounded opeartors between NVS | L3 | |
| 4 | Sep. 10 | Riesz Representation Theorem for Hilbert Spaces | L4, homework 1 due | (RS) II.2 |
| 5 | Sept.15 | Duality, ONB | L5 | (RS) II.2, II.3 |
| 6 | Sept.17 | Convexity: Hahn-Banach (real case) | L6 | (S I.) 5.5 |
| 7 | Sept.22 | Complex Hahn-Banach; Double Dual | homework 2 due, L7 | (S I.) 5.5 |
| 8 | Sept.24 | Topological spaces. Dual topologies | L8 | (R I.) 5.7, (RS) VI.1 |
| 9 | Sept.29 | Baire Category Theorem | L9 | (RS) III.5 |
| 10 | Oct. 1 | Banach-Steinhaus Thm, Open Mapping Theorem | L10 | (RS)III.5 |
| 11 | Oct. 6 | Inverse Mapping Theorem, Lower Bound Theorem, Closed Graph Theorem, | L11 | (RS) III.4 |
| 12 | Oct. 8 | Spectrum of operators. Compact operators. | L12 | (S I.)5.1, (S IV.)3.1, (RS)VI.3 |
| 13 | Oct.13 | Compact Operators (2) | L13, homework 3 due | (R) 4.18, 4.19 |
| 14 | Oct.15 | Compact Operators (3). Weak compactness of closed unit ball in Hilbert spaces. | L14 | (RS)VI.5 |
| 15 | Oct.20 | Spectral Theorem for Compact Operators (1) | L15 | (RS) VI.5 |
| 16 | Oct.22 | SVD for compact operators | L16 | |
| 17 | Oct.27 | Schatten Classes of Compact Operators | L17 | (RS) VI.5 |
| 18 | Oct.29 | Banach Algebras | L18 | (S IV)6.1 |
| 19 | Nov. 3 | Gelfand's Formula | L19, HW4 due (Nov. 4) | (S IV)6.2 |
| 20 | Nov. 5 | Holomorphic Calculus (1) | L20 , pick-up mid-term exam | (S IV)2.3 |
| 21 | Nov. 10 | Holomorphic Calculus (2) | L21, return the mid-term exam, pick HW5 | (S IV)2.3 |
| 22 | Nov. 12 | Holomorphic Calculus (3) | L22 | (S IV) 2.3 |
| 23 | Nov.17 | Fredholm Alternative | L23 | (S IV) 2.3 |
| 24 | Nov.19 | Operators on Hilbert Spaces | L24, HW5 due | (S IV) 2.4 |
| 25 | Nov.24 | Square Root Lemma | L25, pick HW6 | (S IV) 2.4 |
| - | Nov.26 | THANKSGIVING BREAK | NO CLASS | |
| 26 | Dec. 1 | Polar Decomposition | L26 | |
| 27 | Dec. 3 | Spectral Calculus: Resolution of Identity. Continuous Functional Calculus | L27, HW6 due | (S IV) 5 |
| 28 | Dec. 8 | Spectral Calculus: Borel Functional Calculus (1) | L28 | (S IV) 5 |
| 29 | Dec. 10 | Spectral Calculus: Borel Calculus (2) | L29 | (S IV) 5 |
| - | Dec.16 | FINAL EXAM | Online | Online |
