MAT4400 - Autumn 2010

Undervisningsplan

Dato Undervises av Sted Tema Kommentarer / ressurser
24.08.2010 Nadia S. Larsen (NSL)  Aud 2, Vilhelm Bjerknes Hus  Presentation of the course. Metric and topological spaces  § 1.1 
25.08.2010 NSL    Metric and topological spaces, further properties.  §1.1 
31.08.2010 NSL    The Banach space of continuous functions  §1.2 
01.09.2010 NSL    Banach spaces: definition and examples  §1.2 
07.09.2010 NSL    The geometry of Hilbert spaces. Exercises  §1.3. Problems 1.1, 1.6. 
08.09.2010 NSL    Completeness in normed spaces. Bounded operators  § 1.4, § 1.5 
14.09.2010 NSL    Exercises. Sums and quotients of Banach spaces  Problems 1.8, 1.9, 1.10, 1.12 (only for p=2), 1.13, 1.14 
15.09.2010 NSL    Orthonormal bases  § 2.1 
21.09.2010 Sergey Neshveyev    Exercises. The projection lemma and the Riesz lemma.   Problems 1.25 and 1.26. Section §2.2. 
22.09.2010 Sergey Neshveyev    Operators defined via forms. Orthogonal sums. Compact operators  Sections §2.3, §2.4 and §3.1. 
28.09.2010 NSL    Exercises. Compact operators.  Problems 2.3, 2.4, 2.5, 2.6. Section §3.1. 
29.09.2010 NSL    The spectral theorem for compact symmetric operators  §3.2. 
05.10.2010 NSL    The spectral theorem for compact symmetric operators. The Sturm-Liouville problem  § 3.2 -3.3. Problem 3.2 and the following problem: let K(x,y)= cos(x-y) and a=0, b=2 pi in Lemma 3.4. Show that z= pi is the only non-zero eigenvalue. Find the corresponding eigenspace.  
06.10.2010 NSL    Compact operators. Sturm-Liouville operators.  § 3.3-3.4. 
07.10.2010     The obligatory assignment will be posted!  Follow the homepage of the course. 
12.10.2010 NSL    Exercise. Fredholm theory  §3.5. Exercise: use the method of variation of parameters (or variation of constants) to deduce the formula (3.14), in other words use that u+(z,x) and u_(z,x) are solutions for the homogeneous equation to explain why the solution f(x) must be of the form (3.14). 
13.10.2010 NSL    Fredholm theory for compact operators  §3.5. 
19.10.2010 NSL    Exercises. Borel measures. Premeasures  Problem 3.10. Sections §4.1,4.2 
20.10.2010 NSL    Properties of premeasures and measures  §4.2 
26.10.2010 NSL    Extensions via outer measures. Measurable functions.  §4.2, §4.3. 
27.10.2010 NSL    Measurable functions. Integration.  §4.3, §4.4. 
02.11.2010 NSL    Integration. The Lebesgue spaces.   § 4.4, §5.1.  
03.11.2010 NSL    The Lebesgue spaces.  §5.2 
09.11.2010 NSL    L^p is a Banach space  §5.3. Problems 4.7, 4.8, 5.1. 
10.11.2010 NSL    Fourier series in L^2([-\pi, \pi).  Section 1 and the Riemann-Lebesgue lemma in the notes. 
16.11.2010 NSL    Types of convergence for Fourier series. Exercises.  Section 2 and exercises 1-3 in the notes.

In addition, problem 4.9. 
17.11.2010 NSL    Uniform convergence of Fourier series. Exercises.  Section 4 and exercises 4, 5 in the notes. 
23.11.2010 NSL    Product measures. Exercises.  Section 4.5 in Teschl's notes.

Note: this is the last lecture with theory. 
24.11.2010 NSL    Exercises  We look at some of the additional exercises. 
03.12.2010 NSL  Seminar room 313. Time 10:15-12.  Exercises.  This is an extra session where we go over exercises in this list and this list.

I will also answer questions about the material in the course. 
Published Aug. 18, 2010 4:31 PM - Last modified Feb. 27, 2023 11:43 AM
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