| Week | Description | ||||
|---|---|---|---|---|---|
| week 2 | Chapter 1 in Bowers & Kalton | Review of basic notions on normed spaces, show that all norms on Rn are equivalent and prove that compactness is equivalent to bounded closedness in Rn. | |||
| week 3 | banach1.pdf | dual norms in terms of Legendre tranform | |||
| week 4 | banach1.pdf | Geometric Hahn-Banach lemma and Fenchel-Moreau duality theorem, Holder inequality and Fenchel-Rockafellar theorem | |||
| week 5 | Chapter 3 in Bowers & Kalton + banach2.pdf | Relation between geometric Hana-Banach and "extension" Hahn-Banach and applications of general Hahn-Banach | |||
| week 6 | Brezis and Lieb + banach3.pdf | convex analysis explanation of why dual of Lp[0,1] is Lq[0,1] (without using measure theory) | |||
| week 7 | Chapter 4 in Bowers & Kalton | Theorem of Baire, Uniform Boundedness Principle, strong contrapostive formulation of the Uniform Boundedness Principle, Banach-Steihnhaus theorem, Dirichlet kernel | |||
| week 8 | Chapter 4 in Bowers & Kalton | Existence of divergent Fourier series, Open mapping theorem, Bounded inverse theorem, closed range theorem, equivalence of norms theorem, closed maps and closed graph | |||
| week 9 | Chapter 5 in Bowers & Kalton, Vershynin notes | weak and weak* convergence, basics of topological vector spaces from Bowers-Kalton without proofs (Minkowski functional, seminorms, locally convex,…), weak and weak* topology | |||
| week 10 | Chapter 5 in Bowers & Kalton, Vershynin notes | weak and weak* compactness and its sequentially compactness versions, Banach-Alaoglu, norm compactness of unit balls in normed spaces as a characterization of finite-dimensional vector spaces | |||
| week 11 | Chapter 5 in Bowers & Kalton, Vershynin notes | weak and weak* compactness and its sequentially compactness versions, Banach-Alaoglu, norm compactness of unit balls in normed spaces as a characterization of finite-dimensional vector spaces, weak solutions, theorem of Eberlein-Smuylan | |||
| week 12 | Chapter 6 | compact operators, totally bounded sets, basic properties of compact operators, uniform limits of compact operators give compact operators (converse for Hilbert spaces), compactness of multiplication operators on sequence spaces and integral operators with continuous kernels | |||
| week 13 | Chapter 6, 7 | Rank-nullity theorem for compact operators, spectral theorem for compact operators on Banach spaces, and the version for Hilbert spaces, integral operators, Hilbert-Schmidt operators (integral operators), Fredholm alternative, Fredholm operators | |||
| week 14 | Vershynin, Sections on Spectral theory | Spectrum of a bounded operator on a normed space, point, continuous and residual spectrum, resolvent set, Neumann's series, spectrum is compact, spectrum is non-empty, properties of the resolvent operator, spectral radius formula |