Lecturer

Manfred Einsiedler

Coordinator

Andreas Wieser

Sobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity, spectral theory, and unitary representations.

Note that this is the continuation of the course "Functional Analysis I". See this link for the webpage of that course and the topics that were treated.

The first lecture takes place on Moday (18.02.2019).

Time	Place
Monday 10-12	HG G 5
Thursday 13-15	HG G 5

Here is a list of the topics we treated for each week. If not mentioned otherwise, an item was treated with proof. We refer to our main source (see references).

Week 1	Chapter 5.2 up to and including Section 5.2.2.
Week 2	Sections 5.2.3, 5.3.1 and 5.3.2 (where the proof of Lemma 5.50 was not treated yet).
Week 3	Lemma 5.50 and Theorem 5.51. Theorem 6.56, Sections 6.4.1, 6.4.2 and 6.4.4 (where Lemma 6.68 was not yet treated).
Week 4	Finished the proof of Weyl's law and gave some ideas for Section 8.2.2. Section 7.4.5. Multiplication operators on the circle (see e.g. Exercise 6.25).
Week 5	Section 9.1 up to and including 9.1.2. First half of the proof of Theorem 10.1.
Week 6	Choquet's theorem (Prof. Burger).
Week 7	The remaining proof of Theorem 10.1, Proposition 10.2, Lemma 10.3. The Banach algebra of signed measures and L^1-functions on groups.
Week 8	Chapter 11 up to Corollary 11.29.
Week 9	Section 11.3.3, Section 11.4, Section 12.1.1.
Week 10	Sections 12.1.2, 12.1.3, Sections 12.3, 12.5 and 12.6 where not all of the properties (FC1)--(FC6) were discussed. We also treated Section 12.3.4.
Week 11	Continuation of Section 12.6. We began the discussion of projection-valued measures (Section 12.7). Then Section 12.8.1 and 12.8.2.
Week 12	Section 13.1 and 13.2. Then Chapter 14 up to Lemma 14.8.
Week 13	(Taught by Marc Burger) Property (T).
Week 13	Continuation of the proof of the prime number theorem. The Banach algebra inquality was not proven and neither was the Selberg symmetry formula.

Active participation in the exercises classes can yield a bonus of up to 5% on the grade of the exam. The bonus is obtained by presenting exercises in the exercises classes. For the maximal bonus at least 4 points need to be achieved. In this case your bonus will consist in an additional 0.25 at your exam. If you have achieved 2 or 3 points, your mark will be raised by 0.25 only case of a plus-tendence.

The exact procedure is the following:

• The exercise sheet number n is published on this webpage on the Thursday of the (n-1)'th week of the semester.
• You then have until Friday in week n, 12:00 to solve the exercise sheet and volunteer to present one or more exercises on the vorxn-page. Note that you also need to hand in the exercises for which you were selected by Friday in week n, also 12:00. Each assistant has a box in HG J 68 for that purpose. Make sure that you keep a copy of the solutions you need to present for yourself so that you can prepare your talk based on them.
• Per exercise class and exercise one students are selected for a presentation. Which exercise you are supposed to present can be seen in the vorxn shortly after the deadline at 12 o'clock.
• The exercises are then discussed in the exercise class on the Monday of week n+1. It is up to the assistant to decide whether he or she wants your solution presented. The first two times you have been chosen you will be awarded 1 point independently of whether the assistant selected you for presentation (assuming that the solution is correct). To obtain the third point, you must have had one presentation either at that selection or an earlier one. The fourth point will be awarded for a good presentation only.

Exercise sheet	Deadline	Solutions	Topics
Sheet 1	22.02.2019	Solutions for sheet 1	Sobolev spaces on open subsets of Euclidean space.
Sheet 2	01.03.2019	Solutions for sheet 2	Sobolev spaces on open subsets of Euclidean space II.
Sheet 3	08.03.2019	Solutions for sheet 3	Sobolev embedding theorem, (weakly) harmonic functions and the Dirichlet boundary value problem.
Sheet 4	15.03.2019	Solutions for sheet 4	Elliptic regularity, Weyl's law and the heat equation.
Sheet 5	22.03.2019	Solutions for sheet 5	Elliptic regularity at the boundary, Riesz representation and examples of unitary translation operators.
Sheet 6	29.03.2019	Solutions for sheet 6	Spectral theorem for unitary operators.
Sheet 7	05.04.2019	Solutions for sheet 7	Extremal points and Choquet's theorem.
Sheet 8	12.04.2019	Solutions for sheet 8	Haar measures, Banach algebras.
Sheet 9	26.04.2019	Solutions for sheet 9	Unital Banach algebras, spectral radii and Gelfand duals.
Sheet 10	03.05.2019	Solutions for sheet 10	Gelfand transform, Pontryagin dual. Discrete spectrum, approximate point spectrum, approximate spectrum, residual spectrum.
Sheet 11	10.05.2019	Solutions for sheet 11	Spectral theorem for (commuting) normal operators and functional calculus.
Sheet 12	17.05.2019	Solutions for sheet 12	Spectral theorem for unitary representations of abelian groups.

You can enrol for an exercise class here. Note that the first exercise classes will take place in the second week of the semester.

Time	Place	Assistant
Monday 09-10	HG G 26.3	Giuliano Basso
Monday 09-10	HG F 26.5	Emilio Corso

We will be using the book Functional Analysis, Spectral Theory, and Applications by Manfred Einsiedler and Thomas Ward. It is available via SpringerLink here.

Other useful, and recommended references include the following:

• Lecture Notes on "Funktionalanalysis I" by Michael Struwe
• Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
• Elias M. Stein and Rami Shakarchi. Functional analysis (volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011.
Peter D. Lax. Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002.
• Walter Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.