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Undervisningsplan (MAT4340 - Høst 2007)

DatoUndervises avStedTemaKommentarer / ressurser
21.08.2007Nadia S. Larsen  B 70  1.1-1.4  Normerte rom. Kvotientrom. 
22.08.2007Nadia S. Larsen  B 70  1.5, 2.1a - 2.1d  Komplettering av normerte rom. Hilbertrom. 
28.08.2007Nadia S. Larsen  B 70   1.5, 2.1a; Exercises (updated list).   Completion of a normed space; Hilbert spaces; Exercises: chapter 1: 1, 11, 18, 20, 22, 23;. 
29.08.2007Nadia S. Larsen    2.1b - 2.1e; 2.2a - 2.2b  Bessel's inequality, Gram-Schmidt orthogonalisation, Parseval's identity, Projection onto a subspace 
04.09.2007    Sections 2.1e, 2.2a-b. Exercises  Parseval's identity. Distance to a convex set. Projection onto a subspace.

Chapter 2: exercises 1, 3, 5, 8b. 
05.09.2007    Sections 2.2c, 2.3a-b.  Orthogonal decomposition in Hilbert spaces. Linear functionals. 
11.09.2007    Sections 2.3c, 3.1.a. Exercises  Riesz Representation Theorem for a Hilbert space. Examples of linear functionals. The Hahn-Banach theorem and some of its corollaries.

Chapter 2: exercises: 11, 14, 15, 16. 
12.09.2007    Sections 3.1a, 3.2  Corollaries of the Hahn-Banach theorem. Examples of dual spaces. 
18.09.2007NSL    Sections 3.1a, 3.2, 4.1. Exercises  The annihilator of a subspace. Examples of reflexive spaces. Bounded linear operators.

Chapter 3: exercises 2, 3, 4, 5. 
19.09.2007NSL    Sections 4.2, 4.3 a  Examples of linear operators. Compact sets. Compact operators: first properties. 
25.09.2007NSL  B70  Sections 4.2, 4.3a. Exercises  Some examples of linear operators. Compact sets.

Chapter 3, exercise 8. Chapter 4, exercises 1, 2a. 
26.09.2007NSL  B70  Sections 4.3b, 4.4  Compact operators. Dual operators. 
02.10.2007NSL  B70  Section 4.5. Exercises  Operators of finite rank.

Chapter 4, exercises 2b, 5, 6, 7. 
03.09.2007NSL  B70  Sections 4.5a, 4.6, 4.7  Examples of compact operators. Modes of convergence in L(X).  
09.10.2007NSL  B70  Sections 4.5a, 5.1. Exercises  The integral operator in L2. Spectrum of an operator.

Chapter 4: exercises 10, 11, 15, 21, 23. 
10.10.2007NSL  B70  Section 5.2  Fredholm theory for compact operators on Banach spaces 
16.10.2007NSL  B70  Section 5.2, Exercises  Fredholm theory of compact operators

Chapter 4, exercise 9. Chapter 5, exercises 1, 3, 11 
17.10.2007NSL  B70  Section 5.2  Fredholm theory of compact operators 
23.10.2007NSL  B70  Section 6.1-6.2, Exercises  Spectral theory for compact self-adjoint operators

Chapter 5, exercise 9. 
24.10.2007NSL  B70  Section 6.2, 6.3  Spectral theory of compact self-adjoint operators, minimax principle, positive operators. 
30.10.2007NSL  B70  Section 6.2c, 6.3, Exercises  The second Hilbert-Schmidt theorem. Positive operators.

Chapter 6, exercise 1, 3a-c. 
31.10.2007NSL  B70  Section 6.3, 6.4  Ordering in the space of self-adjoint operators. Projections. 
06.11.2007NSL  B70  Exercises. Sections 6.4 and 7.0.   Exercises 1, 3a-c, chapter. 6. Self-adjoint projections. Functional calculus 
07.11.2007NSL  B70  Sections 7.0, 7.1 a.  Functional calculus (continued) 
13.11.2007    No lectures this week   
14.11.2007    No lectures this week   
20.11.2007NSL  B70  Section 7.1b  Spectral integral 
21.11.2007NSL  B70  Sections 7.2, 7.3  Spectral theorem for bounded self-adjoint operators 
27.11.2007NSL  B70  Sections 7.3, 7.4  Spectral family. Spectrum of a self-adjoint operator. 
28.11.2007NSL  B70  Open session (questions), Exercises 1, 2, in chapter 7.