MAT4340 - Autumn 2008

Teaching plan

DateTeacherPlaceTopicLecture notes / comments
25.08.2008Nadia S. Larsen (NL)  B71  Prerequisites (Sections 1.2-1.3)  Review of basic notions from metric spaces. Short overview of the theory of Lebesgue integration.  
26.08.2008NL  B71, 12.15-14.00  Lp-spaces. Normed spaces. (Sections 1.3 and 2.1.)  Lp-spaces. Hölder's and Minkowski's inequalities. Examples of normed spaces. 
01.09.2008NL  B62, 12.15-14  Normed spaces (Sections 2.1 and 2.2)  The spaces L\infty and l\infty. Finite-dimensional normed spaces. Equivalent norms. 
02.09.2008NL  B71, 12.15-14  Banach spaces (Section 2.2)   
08.09.2008NL  B62  Exercises, section 2  Exercises for week 37. 
09.09.2008NL  B 62  Hilbert spaces (Section 3.1)   
15.09.2008NL  B62, 12.15-14  Orthogonality. Orthogonal complements.  Sections 3.2-3.3. 
16.09.2008NL  B71, 12.15-14  Orthonormal bases in Hilbert spaces  Section 3.4 
22.09.2008NL  B62  Exercises. Fourier series  Exercises for week 39. Solution to the additional exercise for week 37 can be found her.

Section 3.5 

23.09.2008NL  B71  Continuous linear transformations  Section 4.1 
29.09.2008NL  B62  Exercises. The norm of a bounded linear transformation  Exercises for week 40.

Section 4.2. 

30.09.2008NL  B71  The space B(X, Y). Inverses of operators.  Sections 4.3-4.4 (Theorem 4.43, Corollary 4.44, Theorem 4.52 will be stated without proofs.) 
06.10.2008NL  B62  Exercises. Inverses of operators. Dual spaces.  Exercises for week 41.

Finish section 4.4.

Section 5.1. 

07.10.2008NL  B71  Seminorms. The second dual. Dual operators.  Section 5.2 (sublinear functionals and seminorms.)

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

Section 5.5.  

13.10.2008NL  B62  Exercises. The adjoint of an operator on Hilbert space.  Exercises for week 42: 5.2, 5.15, example 5.40.

Section 6.1. 

14.10.2008NL  B71  Normal, self-adjoint and unitary operators on Hilbert spaces  Section 6.2. 
20.10.2008NL  B62  Exercises. The spectrum of an operator  Exercises for week 43.

Section 6.3 

21.10.2008NL  B71  The spectrum of an operator. Positive operators. Projections.  Sections 6.3 - 6.4. 
27.10.2008NL  B62  Exercises. Compact operators.  Exercises for week 44.

Section 7.1 

28.10.2008NL  B71  Compact operators. Spectral theory.  Sections 7.1-7.2. 
03.11.2008NL  B62  Exercises. Spectral theory of compact operators.  Exercises for week 45: 7.2, 7.3, 7.8, 7.11. 
04.11.2008NL  B71  Spectral theory of compact operators. The Fredholm alternative.  Section 7.2. 
10.11.2008NL  B62  Self-adjoint compact operators.  Section 7.3. 
11.11.2008NL  B71  Self-adjoint compact operators.  Section 7.3. 
17.11.2008NL  B62  Fredholm integral equations.  Section 8.1. 
18.11.2008NL  B71  Volterra operators.Applications to solving differential equations.  Section 8.2. Section 8.3 (up to Thm. 8.18). 
Published Aug. 25, 2008 4:51 PM - Last modified Oct. 28, 2008 10:47 AM
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