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.
2016-05-12, Franz Luef