| Dato | Undervises av | Sted | Tema | Kommentarer / ressurser |
| 18.01.2010 | Terje Sund (TS)? | B62? | Section 2.2 in "Analysis Now" by G. K. Pedersen? | Baire's Theorem, The Open Mapping Theorem, The Closed Grapf Theorem. The Principle of Uniform Boundedness.Partially and linearly ordered sets. Nets, Zorn's Lemma. Minkowski functionals.Exercises for January 27:E 2.2.1, E 2.2.4, andExercise 1:Let X be the vector space C'[0,1] of all continuously differentiable functions on [0,1] with the sup norm. Let Y = C[0,1], also with the sup norm. Define D: X –> Y by (Dx)(t) = x'(t), for all t in [0,1]. Show that D has closed graph but is not bounded. Explain the result in view of the Closed Graph theorem.Hint: See e.g. Thm. 7.17 in Rudin, Principles of Mathematical Analysis.? |
| 19.01.2010 | TS? | B62? | Section 2.3. Dual spaces. ? | The Hahn-Banach theorem for a real vector space with a given Minkowski functional.? |
| 25.01.2010 | TS? | B62? | Sections 2.3 and 2.4.1-2.4.5 ? | The Hahn-Banach theorem for a complex vector space with a given semi-norm. Consequences of The Hahn-Banach extension theorem for vector spaces. Duality. The adjoint operator. (Topological vector spaces. Weak topologies induced from seminorms.)? |
| 27.01.2010 | TS? | B63? | Sections 2.3.10-11, 2.4.1-2.4.5. Exercises: E.2.1, E 2.4, Exercise 1? | Adjoint operators. Topological vector spaces. Weak topologies induced from seminorms.Comments on E2.2.4? |
| 03.02.2010 | TS? | B63? | Section 2.4.1-2.4.5? | Weak topologies induced from seminorms. The Hahn-Banach separation theorem. (The weak- and weak*-topologies.)? |
| 05.02.2010 | TS? | B62? | Sections 2.4. 2.5. Exercises: E 2.3: 1, 3, 7 (2,4,5)? | More on linear functionals and the weak topology. Minkowski-functionals, seminorm, convex sets, and their relation to topological vector spaces.Comments on E2.3.3? |
| 10.02.2010 | "? | B63? | Section 2.5.? | Minkowski-functionals, seminorms, convex sets, and their relationship to topological vector spaces. Linear functionals on topological vector spaces are open maps. The Hahn-Banach separation theorem.? |
| 12.02.2010 | "? | B62? | Section 2.4.8. Exercises: E 2.4: 1, 2 (,4, 6, 7)? | The weak and the weak-star topology. Alaoglu's Theorem and a corollary to it: The w-star and the norm topology coincide on a normed space X if and only if dim(X) is finite.? |
| 17.02.2010 | "? | B63? | Sections 2.4.10-2.4.12. Section 2.5? | More on the weak and w* toplogies. Annihilators of linear spaces. Proof of Alaoglu's Theorem. (The Krein-Milman Theorem.) ? |
| 19.02.2010 | "? | B62? | Section 2.5 up to 2.5.8(included). Exercises: 2.4: 4, 6,7? | The Krein-Milman Theorem. (Catalogue of extremal boundaries.) E2.4.7 Solution? |
| 24.02.2010 | "? | B63? | 2.5? | The Krein-Milman theorem: The last part of the proof. Catalogue of extremal boundaries: Extreme points in the unit ball of the dual M(X) of C(X), X a compact Hausdorff space. ? |
| 26.02.2010 | "? | B62? | Section 4.1. Exercises: E 2.5:1, 3, [5 b), c), d), (6, 7)]? | Banach algebras ? |
| 03.03.2010 | "? | B63? | Sections 4.2, 4.3? | Unital Banach algebras. Spectrum, spectral radius. ? |
| 05.03.2010 | "? | B62? | Sections 4.1.11, 4.1.12 (4.1.13). Exercises: E 4.1: 3, 4, 9 (not the question about the Volterra operator)? | Holomorphic (analytic) functional analysis. The spectral radius formula. ? |
| 10.03.2010 | "? | B63? | Sections 4.1.13, 4.2? | The spectral radius formula (proof). The Gelfand Transform.? |
| 12.03.2010 | "? | B62? | Sections 4.2, 4.3. Exercises: E2.5.5(c) [and (b)], E 4.1: 10 (,11)? | The Gelfand Transform.? |
| 17.03.2010 | "? | B63? | 4.2: 4-8, 4.3: 1-8? | The Gelfand Transform: Examples.? |
| 19.03.2010 | "? | B62? | Sections 4.3 : 9-12 . Exercises : E 4.1.11, E 4.2: 5,6? | The Stone-Weierstrass Theorem. C*-algebras. ? |
| 24.03.2010 | "? | B63? | 4.3:1-4 ? | The Stone-Weierstrass Theorem and its proof.? |
| 26.03.2010 | "? | B62? | 4.3: 9-19. Exercises: E 4.2: 5, 6, 10; E 4.3: 6.? | C*-algebras? |
| 31.03.2010 | -? | -? | -? | Easter vacation. No lectures on March 31 and April 2.? |
| 07.04.2010 | "? | B63? | 4.3: 14-19, ( 4.4: 1-7)? | Classification of commutative C*-algebras. (Continuous functional calculus, Hilbert's Spectral Theorem - abstract form)? |
| 09.04.2010 | "? | B62? | 4.3.15, Exercises: E 4.2: 10, E 4.3: 6, (9, 12)? | Continuous functional calculus, Hilbert's Spectral Theorem - abstract form? |
| 14.04.2010 | "? | B63? | 4.4? | ? |
| 16.04.2010 | "? | B62? | 4.5. Exercises: E 4.3: 12, 13 (14, 16)? | ? |
| 21.04.2010 | "? | B 63? | 4.5? | The Spectral Theorem for the C*-algebra of all bounded Borel functions on the spectrum of a normal operator.? |
| 23.04.2010 | "? | B 62? | 4.5, Exercises: E 4.3: The remaining part of 12, 16, (14, 9)? | Projection valued measures/spectral measures. ? |
| 28.04.2010 | "? | B 63? | 4.5: 7, 10, 11? | The Spectral Theorem III? |
| 30.04.2010 | "? | B62? | 4.5, 5.1. Exercises: 4.3: 16(remaining part), 14, 9? | Unbounded operators: Domains, extensions, graphs.? |
| 05.05.2010 | "? | B63? | 5.1, (5.2)? | Unbounded operators: Domains, extensions, graphs. (The Cayley Transform.)? |
| 07.05.2010 | TS? | B 62? | 5.2? | The Cayley Transform? |
| 12.05.2010 | TS? | B63? | 5.2? | ? |
| 14.05.2010 | TS? | B62? | 5.2? | ? |
Undervisningsplan
Publisert 29. nov. 2009 14:38
- Sist endret 29. apr. 2011 12:21