Functional Analysis 2 - Winter term 2011/12 (WiSe 2011/12)

Exam results:
You can have a look at your graded test and pick up the Schein at Frau Warcholik's office, B-411, on

WEDNESDAY Feb 8 at 9.30 -- 11.00.

The original test stays with us but you can have a look at it. You can pick up uncollected, marked homework the same time and place. You are not allowed to pick up the Schein for someone else. If it is _completely_ impossible for you to come Wednesday, please make an APPOINTMENT with Frau Warcholik (via phone or email).

Time and place:

Lectures (Prof. Sørensen):
Monday 14:15--16:00, Room C 113. (Starts Oct 17th.)
Tuesday 10:15--12:00, Room C 112.
(There is lecture as usual on Monday Oct 31st!)

Exercise sessions ("Übungen") (A. Michelangeli):
Wednesday 18:15--20.00, Room C 111. (Starts Oct 25th.) Sign up here.

Office hours:
Thomas Østergaard Sørensen (Room B 408): Thursday 10-11.
A. Michelangeli (Room B 334): Wednesday 16-18.

Grader/Korrektor: See the webpage of the Exercises (A. Michelangeli).

Synopsis:
This course is a continuation of the course FA1 from the previous semester (which is, however, no prerequiste - see below). It treats spectral theory of compact, bounded, and un-bounded (mainly self-adjoint) operators, as well as related topics.

For:
Students of Mathematics, Wirtschaftsmathematik, and physics; students in the International Master Programme.
("Gilt für Masterprüfung Mathematik (WP30) und Wirtschaftsmathematik (WP49), Diplomhauptprüfung Mathematik (RM,AM), Diplomhauptprüfung Wirtschaftsmathematik (Kernfach D)").

Prerequisites:
Introductory courses in analysis and linear algebra (Analysis I-III, Lineare Algebra I-II). It is not a prerequisite to have followed FA1 in the past semester, but basic knowledge in Banach and Hilbert-space theory will be needed. (The content of FA1 from last semester is here.) We will also need some complex analysis ("Funktionentheorie"), but this will be treated (briefly) in the course.

Requirement for passing the course:
The course ends with a written final exam (time: To be determined; more details below). The final grade is given based upon the performance on this exam. It is expected that 50% performance will be enough to pass the course, but this number may change slightly. There will be weekly exercise sheets that can be handed in (individually) for grading (in the designated box near the library on the first floor). NO LATE HOMEWORK IS ACCEPTED. (For more details, see the Exercise webpage.) To encourage the regular course work during the semester, the correct solutions to the homework will be counted as bonus points towards the final grade as follows: If one gets 40% or more of the total points on the Exercise Sheets, then the final mark at the Klausur is raised by 0.3/0.4 EXCEPT for marks below 4.0 (and, for 1.0 of course!)

Endklausur (Final (written) Exam):

Exercise Sheets:
Will be posted on the web every Tueday by 14.00 on the exercise homepage. Solutions are due the following Tuesday at 14:00 in the designated box. First sheet is posted on Oct 18th. For more details, see the exercise homepage.

Language:
As in FA1, the lectures, the webpage and our main literature are in English. The purpose is double: to strengthen the English knowledge of the German students and to make the lectures accessible to non-German Master students. The exercise sessions are also held in English, by default. However, Dr Michelangeli and Pr. Sørensen are ready to switch to German in private discussions. If you feel that your English is not strong enough to ask questions, please do it in German. The questions on the Exercise sheets and on the Klausur will be in English, but the solutions can be turned in either in German or in English.

Literature:
As in FA1, there will be no comprehensive Skript (Lecture Notes), since we mainly follow excellent textbooks. The course will not follow a particular textbook. The list below provides a short selection of English and German textbooks on the subject (of which there are many!). Note that most of them cover the material of a two-semester course. The brief contents of the lectures will keep you updated, here you will (sometimes!) find the more precise references.

• G Teschl, Mathematical methods in quantum mechanics, AMS 2009 (online version)
• J Weidmann, Lineare Operatoren in Hilberträumen I + II, Vieweg+Teubner Verlag, 2000 + 2003.
• M Reed and B Simon, Methods of modern Mathematical Physics I: Functional analysis, Academic Press, 1980.
• D Werner, Einführung in die Funktionalanalysis, Springer, 2007.
• H W Alt, Lineare Funktionalanalysis, Springer, 2002.
• N Dunford and J T Schwartz, Linear Operators, Part I--III (pp. 1-2592 (!)), Interscience Publishers, 1972.
• P D Lax, Functional Analysis, Wiley, 2002.
• W Kaballo, Grundkurs Funktionalanalysis, Spektrum Akademischer Verlag, 2011.
• Rudin, Real and Complex Analysis, McGraw and Hill, NY, 1987.
• Rudin, Functional Analysis, McGraw and Hill, NY, 1991.
• M Dobrowolski, Angewandte Funktionalanalysis, Springer, 2006.
• W A Sutherland, Introduction to metric and topological spaces, Oxford University Press, 2009.

Contents (preliminary):

• Motivation and repetition of certains concepts from FA1.
• Spectral Theory for compact operators.
• Spectral Theory for bounded self-adjoint operators.
• Unbounded operators, in particular symmetric operators and quadratic forms.
• Spectral Theory for un-bounded self-adjoint operators.
• Fourier transformation.

Links:

• The MacTutor History of Mathematics archive, University of St. Andrews, Scotland:
  Biographies of all mathematicians (almost ....). See for instance:
  • Banach, Hilbert, (Carl) Neumann, Fredholm, Riesz, Schauder, Dirichlet, Schatten, Schmidt, Bernstein, Weierstrass, Tietze, Urysohn, Weyl, Lebesque, Borel, Dynkin, Lax, Milgram, Herglotz, Nevanlinna, Riemann, Stieltjes, Hausdorff, Dirac, Abel, Haar, Gelfand, Mazur, Cauchy, Dunford, (Jacob) Schwartz,