| Week | Topic | Exercises | Comments |
|---|---|---|---|
| 2 | Banach spaces, operators, and dual spaces. An introduction to $\ell^p$-spaces. Read Chapter 1, and the beginning of 2.1. Revise A.3 and basic point-set topology (compactness in metric and topological spaces in particular). | 1.6, 1.7, 1.8, 1.12, 1.14, 2.1, 2.8, 2.9, 2.10 | |
| 3 | Duality of $\ell^p$-spaces. A recap of measure theory. $L^p$-spaces. The Radon-Nikodym theorem. The Riesz representation theorem (without proof). Read 2.1-2.3, and A.4-A.5. | 2.5, 2.6, 2.10, 2.11, 2.12, 2.13, 2.14 | |
| 4 | The Hahn-Banach theorem. The invariant Hahn-Banach theorem. Banach limits. Haar measures. Read 3.1-3.4. | 3.4, 3.9, 3.10, 3.11, 3.16, 3.17, 3.18 | |
| 5 | Applications of Hahn-Banach: Dual and bidual spaces. Reflexive spaces. The adjoint of an operator. Direct sums and quotients of Banach spaces. Read 3.5-3.8. | 3.1, 3.2, 3.3, 3.7, 3.8 | |
| 6 | Direct sums and quotients of Banach spaces. The Baire category theorem. Read 3.7 - 3.8 and 4.1 | 4.1, 4.4, 4.9 | |
| 7 | The uniform boundedness principle. Applications to Fourier series and weak analyticity. Read 4.2. | 4.10, 4.11, 4.12 | |
| 8 | The open mapping theorem. Read 4.3. No lecture on Friday 24.2 | ||
| 9 | The closed graph theorem. Applications of open mapping and closed graph theorems. A crash course in topology. Read 4.3, 4.4, 5.1. | 4.18, 4.19, 4.21, 5.1, 5.2, 5.5, 5.9 | |
| 10 | Topological vector spaces and their dual spaces. Examples. Read 5.2-5.3. | 5.11, 5.12, 5.13, 5.14 | |
| 11 | Locally convex topological vector spaces, the Hahn-Banach theorem, semi-norms. The weak and weak* topologies. Weak and weak* convergence. Read 5.4. | 5.17, 5.19 5.20, 5.21 | Ekstra resurs: Harald Hanche-Olsens notater. |
| 12 | The weak and weak* topologies. Weak and weak* convergence. The Banach-Alaouglu theorem. Application: the Herglotz representation. Read 5.4-5.5. | 5.24, 5.25, 5.26, 5.31, 5.33 | Ekstra resurs: Harald Hanche-Olsens notater. |
| 13 | Mazur's, Goldstine's, and Kakutani's theorems. Compact operators: definition. Read 6.1, and Schauder's theorem in 6.2. Friday lecture is cancelled. | 6.1, 6.2, 6.3, 6.4, 6.5 | |
| 14 | Easter: No lectures. | ||
| 15 | General spectral theory for operators. Read 8.1. | 8.4, 8.10, 8.13 | |
| 16 | Classification of the spectrum of an operator. Spectral theory of compact operators. The spectral theorem of compact symmetric operators on a Hilbert space. Read 8.1 and 6.2. | 6.6, 6.7, 6.9, 6.12 | The spectral theorem for compact symmetric operators is Theorem 7.30 in the book. |