Functional Analysis - Summer term 2011 (SoSe 2011)

Make-up exam / Wiederholungsklausur:

Result of the Make-up exam

Exercise sessions ("Übungen") (PD Oliver Matte):
Monday 16:15--18.00, B 051. (Starts May 9th.)

Change last week: Lectures: Monday July 25th, 16:15--18.00, Wednesday July 27th, 8:15--10:00. Exercise sessions Friday July 29th, 8:15--10:00 ("Klausur-vorrechnung").

Office hours:
Thomas Østergaard Sørensen (Room B 408): Thursday, 10:00--11:00
Oliver Matte (Room B 404): Monday, 14:00--15:00
Torben Krüger (Room B 208): Thursday, 10:00--11:00

Graders/Korrektoren: See the webpage of the Exercises (PD Matte).

Synopsis:
Functional analysis can be viewed as ``linear algebra on infinite-dimensional vector spaces'', where these spaces (often) are sets of functions. As such it is a merger of analysis and linear algebra. The concepts and results of functional analysis are important to a number of other mathematical disciplines, e.g., numerical mathematics, approximation theory, partial differential equations, and also to stochastics; not to mention that the mathematical foundations of quantum physics rely entirely on functional analysis. This course will present the standard introductory material to functional analysis (Banach and Hilbert spaces, dual spaces, Hahn-Banach Thm., Baire Thm., Open Mapping Thm., Closed Graph Thm.). We will also cover Fredholm theory for compact operators and the spectral theorem. These are powerful tools for applications to PDE's and quantum mechanics, respectively.

For:
Students of Mathematics, Wirtschaftsmathematik, and physics; students in the International Master Programme.
("Gilt für Bachelorprüfungen Mathematik (WP4) und Wirtschaftsmathematik (P12), Masterprüfung Wirtschaftsmathematik (WP11), Diplomhauptprüfung Mathematik (RM,AM), Diplomhauptprüfung Wirtschaftsmathematik (Kernfach D)").

Prerequisites:
Introductory courses in analysis and linear algebra (Analysis I-III, Lineare Algebra I-II).

Requirement for passing the course:
The course ends with a written final exam (time: Friday, July 22nd, 15:00--17:30; 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 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 Monday by 12.00 noon on the exercise homepage. Solutions are due the following Monday at 12:00 noon in the designated box. First sheet is posted on May 9th. For more details, see the exercise homepage.

Language:
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, Herr Krüger, Dr. Matte, 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:
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 find the more precise references.

• M Reed and B Simon, Methods of modern Mathematical Physics I: Functional analysis, Academic Press, 1980.
• D Werner, Einführung in die Funktionalanalysis, Springer, 2007.
• 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
• Topological and metric spaces. (Topological spaces: basics; continuity and convergence; metric spaces; example: sequence spaces; compactness; example: space of continuous functions; Baire's theorem).
• Banach and Hilbert spaces. (Vector spaces; Banach spaces; linear operators; linear functionals and dual space; Hilbert spaces).
• Measures, integration and Lp-spaces. (Measures; integration; Lp-spaces; decomposition of measures).
• The cornerstones of functional analysis. (Hahn-Banach extension theorem; three consequences of Baire's theorem; (bi-)dual space and weak topologies; bounded operators).
• Topologies on bounded operators. (Adjoint operator; the spectrum; compact operators; Fredholm alternative and canonical form for compact operators).

Links:

• The MacTutor History of Mathematics archive, University of St. Andrews, Scotland:
  Biographies of all mathematicians (almost ....). See for instance:
  • Hausdorff, Bolzano, Weierstrass, Cauchy, H"older, Minkowski, Banach, Riesz, Schwarz, Hilbert, Frechet, Hamel, Zorn, Fourier, Pythagoras, Bessel, Parseval, Lebesgue, Fischer, Beppo-Levi, Hahn, Steinhaus, Alaoglu,