| Date | Teacher | Place | Topic | Lecture notes / comments |
| 19.01.2011 | Terje Sund (TS)? | B71? | Section 2.2 in "Analysis Now" by G. K. Pedersen? | 15:15-17:00 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 28: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 the derivation operator 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. ? |
| 21.01.2011 | TS? | B81 NB !!? | Section 2.3? | 14:15 -16:00 The Closed Graph Theorem and the Uniform Boundedness Principle. The Hahn-Banach Theorem for a real vector space with a given Minkowski functional. ? |
| 26.01.2011 | TS? | B71? | 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.) ? |
| 28.01.2011 | TS? | B63? | Sections 2.3.10-11, 2.4.1-2.4.5. Exercises: E.2.1, E 2.4, Exercise 1 (see January 19)? | Adjoint operators. Topological vector spaces. Weak topologies induced from seminorms.? |
| 02.02.2011 | TS? | B71? | Section 2.4.1-2.4.5? | Weak topologies induced from seminorms. The Hahn-Banach separation theorem. (The weak- and weak*-topologies.) ? |
| 04.02.2011 | TS? | B63? | Sections 2.4. 2.5. Exercises: E 2.3: 1, 3, 7 (2,4,5) ? | Kommentar til E 2.2.4More on linear functionals and the weak topology. Minkowski-functionals, seminorm, convex sets, and their relation to topological vector spaces.? |
| 09.02.2011 | TS? | B71? | 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. ? |
| 11.02.2011 | TS? | B63? | 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.? |
| 16.02.2011 | TS? | B71? | 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.) ? |
| 18.02.2011 | TS? | B63? | 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 ? |
| 23.02.2011 | TS? | B71? | 2.5? | Catalogue of extremal boundaries. Extreme points in the unit ball of the dual M(X) of C(X), X a compact Hausdorff space.? |
| 25.02.2011 | TS? | B63? | Section 4.1. Exercises: E 2.5:1, 3, [5 b), c), d), (6, 7)]? | Extreme points in the unit ball of the dual M(X) of C(X), X a compact Hausdorff space. ? |
| 02.03.2011 | TS? | B71? | Section 4.1? | Banach algebras. Unital Banach algebras. (Spectrum, spectral radius) ? |
| 04.03.2011 | TS? | B63? | 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. ? |
| 09.03.2011 | TS? | B71? | Sections 4.1.13, 4.2 ? | The spectral radius formula (proof). The Gelfand Transform. ? |
| 11.03.2011 | TS? | B63? | Sections 4.2, 4.3. Exercises: E2.5.5(c) [and (b)], E 4.1: 10 (,11)? | The Gelfand Transform. ?? |
| 16.03.2011 | TS? | B71? | 4.2: 4-8, 4.3: 1-8 ? | The Gelfand Transform. Examples. ? |
| 18.03.2011 | TS? | B63? | Sections 4.2 : 4-8 . Exercises: E4.1.11, E 4.2: 5,6 ? | The Gelfand Transform: Examples.? |
| 23.03.2011 | TS? | B71? | 4.3? | The Stone-Weierstrass Theorem. C*-algebras.? |
| 25.03.2011 | TS? | B63? | 4.3: 1-4. Exercises: E 4.2: 5, 6, 10; E 4.3: 6. ? | The Stone-Weierstrass Theorem and its proof. ? |
| 30.03.2011 | TS? | B71? | 4.3: 9-19.? | C*-algebras? |
| 01.04.2011 | TS? | B63? | 4.3: 14-19, ( 4.4: 1-7) Exercises: E4.2: 10, E 4.3: 6, (9, 12) ? | Classification of commutative C*-algebras. (Continuous functional calculus, Hilbert's Spectral Theorem - abstract form) ? |
| 06.04.2011 | TS? | B71? | 4.3.15? | Continuous functional calculus, Hilbert's Spectral Theorem - abstract form ? |
| 08.04.2011 | TS? | B63? | 4.4 Exercises: E 4.3: 12, 13 (14, 16) ? | ? |
| 11.04.2011 | TS? | B534 NB! 1215-14? | 4.5. ? | The Spectral Theorem for the C*-algebra of all bounded Borel functions on the spectrum of a normal operator. ? |
| 13.04.2011 | TS? | B534 NB! 1215-14? | 4.5. Exercises: E 4.3: 16, 14 (, 9)? | The Spectral Theorem for the C*-algebra of all bounded Borel functions on the spectrum of a normal operator. Projection valued measures/spectral measures. ? |
| 20.04.2011 | -? | -? | -? | P?skeferie. Ingen undervisning 20. og 22. april.? |
| 27.04.2011 | TS? | B71? | 4.5: 7, 10, 11 ? | The Spectral Theorem III ? |
| 29.04.2011 | TS? | B63? | 4.5, 5.1. Exercises: E 4.3: 16(remaining part), 14, 9? | Unbounded operators: Domains, extensions, graphs. ? |
| 03.05.2011 | TS? | B71? | 5.1, (5.2) ? | Unbounded operators: Domains, extensions, graphs. (The Cayley Transform.) ? |
| 05.05.2011 | TS? | B63? | 5.2? | The Cayley Transform ? |
| 10.05.2011 | TS? | B71? | 5.2? | ? |
| 12.05.2011 | TS? | B63? | 5.2? | ? |
Teaching plan
Published Jan. 12, 2011 1:42 PM
- Last modified Apr. 29, 2011 12:24 PM