| Week | Description | |
|---|---|---|
| week 2 | Chapter 1 in Bowers & Kalton | Review of basic notions on metric spaces and in particular normed spaces, space of bounded linear operators, sequence spaces: Basic properties (normed space, complete.) Hoelder and Minkowski inequalities for sequences |
| week 3 | Chapter 2 in Bowers & Kalton | Section 2.1. and the criterion for completeness of normed spaces in 2.3. Duality of sequence spaces. Zorn's lemma K. Conrad's notes on Zorn's leamma How to use Zorn's lemma by Tim Gowers |
| week 4 | Chapter 3 in Bowers & Kalton | Theorem of Hahn-Banach with proof from Vershynin's notes, Sections 3.1-3.3. Continue the discussion of Hahn-Banach for sublinear functionals, Consequences of Hahn-Banach, Section 3.3, 3.6 |
| week 5 | Chapter 3 in Bowers & Kalton | natural embedding, the adjoint of an operator, Volterra operator (Section 3.6.), Construction of new Banach spaces from old (Section 3.7.) Direct sum, quotient space, Calkin algebra |
| week 6 | Chapter 4 in Bowers & Kalton | Baire category theorem and some of its variants as described in the book |
| week 7 | Chapter 4 in Bowers & Kalton | Open mappging theorem, closed graph theorem, uniform boundedness principle, Banach-Steinhaus theorem, divergent Fourier series via Banach-Steinhaus, Hellinger-Toeplitz theorem Supplement |
| week 8 | Chapter 4 in Bowers & Kalton | Bounded inverse theorem, Riemann-Lebesgue Lemma, Closed Range Theorem (not in the book). |
| week 9 | Chapter 4 and 5 in Bowers & Kalton | Projections in Banach spaces, complemented subspaces, Basic notions of topology and topological vector, weak topology spaces |
| week 10 | Chapter 5 in Bowers & Kalton | weak convergence, weak-* topology and weak-* convergence, examples for sequence spaces, Lemma of Riesz on proper closed subspaces of Banach spaces, non-compactness of the unit ball in the norm topology of a Banach space, Banach-Alaoglu, uniform, strong and weak convergence for operators |
| week 11 | Chapter 6 in Bowers & Kalton | Weierstrass approximation theorem, compactness in metric spaces, compact operators |
| week 12 | Easter break | |
| week 13 | Chapter 6 in Bowers & Kalton | Compact operators, ideal property, examples |
| week 16 | Chapter 6,7 in Bowers & Kalton | Schauder's Theorem for compact operators, compact perturbation of the identity, rank-nullity theorem, Hilbert-Schmidt operators, Theorem 7.30 spectral theorem for selfadjoint compact operators |
| week 17 | Chapter 8 in Bowers & Kalton | Banach algebras, spectrum, material in Section 8.1. |