MAT4350 - Spring 2009

Undervisningsplan

DatoUndervises avStedTemaKommentarer / ressurser
12.01.2009Nadia S. Larsen (NL)? B 62? The open mapping theorem. The closed graph theorem. The principle of uniform boundedness? Section 2.2 in Pedersen's book "Analysis Now".

Exercises for January 19: E 2.2.4 and E 2.2.6 in "Analysis Now".?

16.01.2009NL? B62? The principle of uniform boundedness (continued). Brief synopsis of orderings on a set.? We finish section 2.2 in "Analysis now" and expand the hints to prove E. 2.2.4 and 2.2.6.

For several purposes later we need a tool from set theory, namely Zorn's lemma, and we shall review the notation and concepts needed for its formulation (sections 1.1.1-1.1.3).?

19.01.2009NL? B62. NOTE TIME: 9:15-10. We take 2 hours in the afternoon.? Minkowski functionals on real vector spaces. Fundamental lemma.? We start on section 2.3 in "Analysis Now". ?
19.01.2009NL? B62. NOTE TIME: 14:15-16:00? The Hahn-Banach extension theorem for vector spaces. Consequences. Duality ? We continue with section 2.3.?
23.01.2009NL? B62? Exercises E. 2.2.4 and 2.2.6.? Exercises for Friday, January 30: E 2.3.1, E 2.3.2 (see also exercise 5.3 and Corollary 5.49 in Rynne-Youngson), E 2.3.4, E 2.3.5.?
26.01.2009NL? B62? The adjoint operator. Topological vector spaces. Weak topologies induced from seminorms.? Section 2.3. Sections 2.4.1-2.4.5.?
30.01.2009NL? B62? Exercises.? Exercises E 2.3.1, E 2.3.2 (see also exercise 5.3 and Corollary 5.49 in Rynne-Youngson), E 2.3.4, E 2.3.5.

Exercises for Friday, February 6: E 2.3.3, E 2.4.7.?

02.02.2009NL? B62? The Hahn-Banach separation theorem. The weak and weak*-topologies. ? Section 2.4.?
06.02.2009NL? B62? Exercises E. 2.3.3, E.2.4.7.? ?
09.02.2009NL? B62? The weak and weak*-topologies. Alaoglu's theorem. ? From "Analysis Now": 2.3.6, 2.4.8-2.4.12 and 2.5.1-2.5.4.?
13.02.2009NL? B62? Exercises.? Exercises E 2.3.6, E. 2.4.1, E. 2.4.4, E. 2.4.6.?
16.02.2009NL? B62? Extremal boundary. The Krein-Milman theorem. Catalogue of extremal boundaries.? Sections 2.5.3-2.5.7.?
20.02.2009NL? B62? Exercises? E 2.5.5 (a) and (b). E 2.5.7.?
23.02.2009NL? B62? Banach algebras. Spectrum. Resolvent set. Spectral radius.? Sections 4.1.1.-4.1.12.?
27.02.2009NL? B62? Exercises? E 4.1.1, E 4.1.7.?
02.03.2009NL? B62? The spectrum of an element in a Banach algebra. Ideals and characters.? Sections 4.1.10-4.1.14 and 4.1.2, 4.2.1.?
06.03.2009NL? B62? Exercises? E. 4.1.3, 4.1.8, 4.1.13.?
09.03.2009NL? B62? The Gelfand transform. Examples.? Sections 4.2.1-4.2.7.?
13.03.2009NL? B62? Exercises? E. 4.1.9, 4.2.1, 4.2.2.?
16.03.2009NL? B62? The Gelfand transform on L^1(R). The Stone-Weierstrass theorem.? Sections 4.2.8 and 4.3.1-4.3.5.?
20.03.2009NL? B62? Exercises? E 4.1.9 and E.4.2.5, 4.2.6, 4.2.11, 4.2.13.In exercise 4.2.5 by the generation property one means that the set of polynomials P(I, A1,..., An) in the variables I, A1, ..., An is dense in the given (commutative) Banach algebra.?
23.03.2009Simen Rustad? B62? Involutions. C*-algebras. Normal, self-adjoint, unitary operators. ? Sections 4.3.7-4.3.12.?
27.03.2009? ? No lecture today!? ?
30.03.2009NL? B62? Spectral theory for commutative C*-algebras? Sections 4.3.14-4.3.19.?
03.04.2009NL? B62? Spectral theorem, spectrum and eigenvalues.? Sections 4.4.1-4.4.7.?
06.04.2009? ? Easter-week. No lecture today.? ?
10.04.2009? ? Easter week. No lecture today.? ?
13.04.2009? ? Easter. No lecture today.? ?
17.04.2009NL? B62? Exercises? E 4.3.1, 4.3.6, 4.4.4, 4.4.5. ?
20.04.2009NL? B62? Spectral theorem with Borel function calculus? Section 4.5.?
24.04.2009NL? B62? Spectral theorem with Borel function calculus? Section 4.5.?
27.04.2009NL? B62? Unbounded operators. Domains, extensions, graphs.? Section 5.1.?
01.05.2009? ? No lecture today.? ?
04.05.2009NL? B62? Unbounded operators. The Cayley transform.? Sections 5.1 (without 5.1.10-5.1.11 and 5.1.13) and 5.2.1-4.?
08.05.2009NL? B62? The Cayley transform? Sections 5.2.4-6.?
Published Jan. 5, 2009 2:56 PM - Last modified Apr. 24, 2009 2:06 PM
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