MAT4340 - Autumn 2009

Undervisningsplan

DatoUndervises avStedTemaKommentarer / ressurser
25.08.2009Terje Sund (TS)? B71? Sections 1.2-1.3? Short review of basic notions from metric spaces and of the theory of Lebesgue integration.?
31.08.2009TS? B62? Sections 1.3 and 2.1. Lp-spaces. Normed spaces.? Lp-spaces. H?lder's and Minkowski's inequalities. Examples of normed spaces. ?
01.09.2009TS? B71? Sections 2.1 and 2.2. Normed spaces.? The spaces L∞ and l∞. Finite-dimensional normed spaces. Equivalent norms. ?
07.09.2009TS? B62? Section 2.3. Banach spaces.? Riesz' Lemma. When is the unit sphere compact? Convergence versus absolute convergence of infinite series.?
08.09.2009TS? B71? Exercises for Section 2? Problem set 1

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14.09.2009 "? B62? 3. Hilbert spaces? Sections 3.1, 3.2?
15.09.2009 "? B71? Orthogonality. Orthogonal complements. ? Sections 3.2-3.3. ?
21.09.2009 "? B62? Orthonormal bases in Hilbert spaces? Section 3.4?
22.09.2009"? B71? Exercises. Fourier series? Problem set 2

Section 3.5

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28.09.2009"? B62? Continuous linear transformations ? Section 4.1 ?
29.09.2009 "? B71? Exercises. The norm of a bounded linear transformation? Problem set 3.

Section 4.2

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05.10.2009 "? B62? The Space B(X,Y). The inverse of an operator.? Sections 4.3 and 4.4 (Thm. 4.43, Cor. 4.44, and Thm. 4.52 without proofs.)?
06.10.2009"? B71? Exercises. The inverse of an operator. Dual spaces.? Problem set 4:

Exercises 4.6, 4.7, 4.11, 4.14, 4.17, and the following Problem

The rest of Section 4.4.

Section 5.1?

12.10.2009"? B62? The dual of l^p. Extensions of functionals defined on subspaces.? The last part of Section 5.1

Section 5.3 (The Hahn-Banach extension theorem in normed spaces, Thm. 5.19, without proof.)

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13.10.2009 "? B71? Exercises. Dual operators. The adjoints of operators on Hilbert spaces.? Problem set 5:

Exercise 5.2. The solution of (b) given in the book is wrong. Find the error and give a correct solution.

(Problem. Prove that the Banach space l^p is reflexive for 1<p<∞ .)

Section 5.5, Example 5.40. Section 6.1?

19.10.2009 "? B62? Normal, self-adjoint, and unitary operators on Hilbert spaces? Section 6.1, Section 6.2?
20.10.2009 "? B71? Exercises. The spectrum of an operator.? Problem set 6:

Problem 1. Prove that the Banach space l^p is reflexive for 1<p<∞ .

Exercises 6.1, 6.3, 6.7 (only for the exercise in 6.1), 6.10, 6.13, 6.15, and the following

Problem

Section 6.2. Section 6.3?

26.10.2009 "? B62? The spectra of normal and self-adjoint operators? Section 6.3 ?
27.10.2009 "? B71? Exercises. Orthogonal projections. Functions of self-adjoint operators? [Problem set 7: Exercises 6.18, 6.22, 6.27, 6.28, and the following Problem will be dicussed on November 3.]

Section 6.4 ?

02.11.2009"? B62? Positive square roots. Compact operators? Section 6.4 and Section 7.1?
03.11.2009 "? B71? Exercises. Compact operators? Problem set 7: Exercises 6.18, 6.22, 6.27, 6.28, and the following

Problem

Sections 7.1 and 7.2?

09.11.2009"? B62? Spectral Theory of Compact Operators. ? Section 7.2 ?
10.11.2009"? B71? Exercises. Section 7.2? Problem set 8: Exercises 7.2, 6.5, 7.6, 7.7, 7.10, 7.11

(Examples 4.7 and 4.41)?

16.11.2009TS? B62? Section 7.3? Self adjoint compact operators.?
17.11.2009TS? B71? Section 7.3? Problem set 9: Exercises 7.8, 7.16, 7.17, 7.18, 7.20, 7.22, 7.23.?
Published Aug. 19, 2009 2:42 PM - Last modified Sep. 3, 2010 12:37 AM
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