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Undervisningsplan

DatoUndervises avStedTemaKommentarer / ressurser
16.11.2006    7.3  Spectral family and spectrum of self-adjoint operators. (7.4 Simple spectrum, will not be included in the pensum) Exercises: Chap. 7: 1, 2 (, 3) 
14.11.2006    7.2, 7.3  Hilbert´s theorem. Spectral decomposition of selfadjoint (bounded) operators. 
09.11.2006    7.1, 7.2  Spectral decomposition. Spectral integral. Hilbert´s theorem.

Exercises: 7.04, page 107 

07.11.2006    7.0, 6.4  Functional calculus. Orthoprojections. 
02.11.2006    6.3, 6.4a)  Ordering. Projections in linear spaces.

Exercises: Chap.6: 7, 11, 12 

31.10.2006    6.3  Ordering in the space of self-adjoint operators. 
26.10.2006    6.2c), 6.3   The second Hilbert-Schmidt Theorem. (Ordering in the space of self-adjoint operators.)

Exercises: Chap.6: 5, and in addition:

Exercise. Let K be an integral operator on H=L_2[a,b] with a continuous kernel function k. Assume that K has a Riemann square-integrable eigenfunction f in H with a nonzero eigenvalue. Prove that

(a) f is continuous

(b) If k has continuous partial derivatives of order 1 (respectively n) then f is continuously differentiable (resp. n times continuously differentiable). 

24.10.2006    6.2 b, c   The Minimax principle. The second Hilbert-Schmidt theorem.  
19.10.2006    6.2a  The first Hilbert-Schmidt theorem.

Exercises: Chap. 6: 2, 3 

17.10.2006    6.1, 6.2a  Spectral theory for compact self-adjoint operators. 
12.10.2006    5.2, 4.4, 6.1  Fredholm´s theory of compact operators. Adjoint operators.

Exercises: Chap. 4: 23; Chap.5: 9, 11 

10.10.2006    5.2  Compact operators. Fredholm´s theory. 
05.10.2006    5.2  Continuous spectrum. Compact operators.

Exercises: Chap. 5: 3, 4, 5, and in addition:

Let X and Y be normed spaces, Y complete, D a dense subspace of X. Assume A : D -->Y is a bounded linear operator. Prove that A has an extension to an operator in L(X,Y) with the same norm as A. Also show that this extension of A is unique. 

03.10.2006    4.7, 5.1, 5.2  Invertible operators. The spectral theory of compact operators: Point spectrum. 
28.09.2006    4.6  K(X,Y) is closed in L(X,Y) : completion of proof (there is a gap at the end of the argument given in the book). Different types of operator convergences. If there is time: A in L(X,Y) compact implies the adjoint (dual operator) A* is compact.

Exercises: Chap. 4: 8, 12, 20 

26.09.2006    4.4, 4.5a  Approximation of compact operators by operators of finite rank. Dual operators. 
21.09.2006    4.5  Finite rank operators.

Exercises: Chap. 4: 1, 3, 15, (19: uses exercise 8 of Chap.3) 

19.09.2006    4.2, 4.3a  Examples of bounded linear operators. Precompact sets. Compact operators. 
14.09.2006    3.1, 4.1  Corollaries to the Hahn-Banach theorem. Bounded linear operators.

Exercises: Chap. 2: 24, Chap. 3: 5 

12.09.2006    3.1, 3.2  The Hahn-Banach theorem. Examples of dual spaces. 
07.09.2006    2.2 c, 2.3 a-c, 3.2  Linear functionals. Dual spaces. Examples.

Exercises: Chap. 2: 10, 11, 12, 13, 29 

05.09.2006    2.1 e, 2.2 a, b, c  Application of Parseval´s identity to L_2[a,b]. Convex sets. Orthogonal projections, orthogonal complements. 
31.08.2006    2.1 e), 2.2 b)  Anvendelse av Parsevals identitet på L_2[a,b]. Konvekse mengder.

Oppgaver.

Kap.1: 18, 20

Kap.2: 1, 4, 5, 8 c). 

29.08.2006    2.1 b)- 2.1 e)  Bessels ulikhet og Parsevals identitet. Eksistens av ortonormal basis. 
24.08.2006    1.5, 2.1 a)  Komplettering av normerte rom. Hilbertrom.

Oppgaver.

Kap.1: 1, 11, 12, 22, 23,24.

Dessuten:

(I) Vis at rommet Rp[a,b] (side 16) ikke er et lineært rom. Forsøk å omdefinere Rp[a,b] slik at det blir lineært (og fremdeles inneholder C[a,b]). 

22.08.2006Terje Sund  B 70  1.1 - 1.4.   Normerte rom. Kvotientrom. 
Publisert 14. aug. 2006 16:26 - Sist endret 14. nov. 2006 14:40