Teaching plan

DateTeacherPlaceTopicLecture notes / comments
19.01.2011Terje 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, and

Exercise 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. ?

21.01.2011TS? 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. ?
26.01.2011TS? 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.) ?
28.01.2011TS? 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.?
02.02.2011TS? B71? Section 2.4.1-2.4.5? Weak topologies induced from seminorms. The Hahn-Banach separation theorem. (The weak- and weak*-topologies.) ?
04.02.2011TS? B63? Sections 2.4. 2.5. Exercises: E 2.3: 1, 3, 7 (2,4,5) ? Kommentar til E 2.2.4

More on linear functionals and the weak topology. Minkowski-functionals, seminorm, convex sets, and their relation to topological vector spaces.?

09.02.2011TS? 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. ?
11.02.2011TS? 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.?
16.02.2011TS? 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.) ?
18.02.2011TS? 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 ?
23.02.2011TS? 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.?
25.02.2011TS? 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. ?
02.03.2011TS? B71? Section 4.1? Banach algebras. Unital Banach algebras. (Spectrum, spectral radius) ?
04.03.2011TS? 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. ?
09.03.2011TS? B71? Sections 4.1.13, 4.2 ? The spectral radius formula (proof). The Gelfand Transform.

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11.03.2011TS? B63? Sections 4.2, 4.3. Exercises: E2.5.5(c) [and (b)], E 4.1: 10 (,11)? The Gelfand Transform.

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16.03.2011TS? B71? 4.2: 4-8, 4.3: 1-8 ? The Gelfand Transform. Examples. ?
18.03.2011TS? B63? Sections 4.2 : 4-8 . Exercises: E4.1.11, E 4.2: 5,6 ? The Gelfand Transform: Examples.?
23.03.2011TS? B71? 4.3? The Stone-Weierstrass Theorem. C*-algebras.?
25.03.2011TS? B63? 4.3: 1-4. Exercises: E 4.2: 5, 6, 10; E 4.3: 6. ? The Stone-Weierstrass Theorem and its proof. ?
30.03.2011TS? B71? 4.3: 9-19.? C*-algebras?
01.04.2011TS? 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) ?
06.04.2011TS? B71? 4.3.15? Continuous functional calculus, Hilbert's Spectral Theorem - abstract form ?
08.04.2011TS? B63? 4.4 Exercises: E 4.3: 12, 13 (14, 16) ? ?
11.04.2011TS? 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. ?
13.04.2011TS? 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.?
27.04.2011TS? B71? 4.5: 7, 10, 11 ? The Spectral Theorem III ?
29.04.2011TS? B63? 4.5, 5.1. Exercises: E 4.3: 16(remaining part), 14, 9? Unbounded operators: Domains, extensions, graphs. ?
03.05.2011TS? B71? 5.1, (5.2) ? Unbounded operators: Domains, extensions, graphs. (The Cayley Transform.) ?
05.05.2011TS? B63? 5.2? The Cayley Transform ?
10.05.2011TS? B71? 5.2? ?
12.05.2011TS? B63? 5.2? ?
Published Jan. 12, 2011 1:42 PM - Last modified Apr. 29, 2011 12:24 PM