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

## Time and place:

Exercise sessions ("Übungen") (A. Michelangeli):

Tutorium/repetitorium sessions (A. Michelangeli):
Wednesday 14:15--16:00, Room B 132. (Starts Oct 18th with 'Warm Up Tutorial'.)

(For organisation and programme for the exercises, and tutorium, see the link above. )

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).

## General information for the course:

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):

Time: Saturday Feb 4th, 09:00 (duration: 120 min)
Place: Room B-132

Material: Complete material of all lectures up until and including Tuesday Jan 31th, and Exercise sheets up until and including Sheet 13. The solutions to Sheet 13 will be posted on Wed Feb 1st.

You are allowed to bring 1 (one) two-sided HANDWRITTEN cheat sheet (Spickzettel): a piece of A4-paper on which you have written (on both sides) whatever you think might be useful for the Klausur/exam. Anyone with a (partially or entirely) printed/photo copied cheat sheet will be expelled from the exam. You may not bring anything else (apart from pens/pencils), i.e., no books, notes from class, homework, solutions to homework etc. Examination booklet and extra paper will be provided. Put your name on every sheet you wish to hand in, and write readable.

There will be more problems than you need to solve to get the maximal point (100%), so you will have some freedom to choose, but you can attempt all of them and collect partial credits. 50% performance will be enough to pass the course.

The final grade is determined by the final exam plus the bonus points from homework, see the precise rules above.

There will be no make-up exam (keine Nachholklausur) and there will be no other exam (keine Wiederholungsklausur) offered.

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.