Vorlesung: Funktionalanalysis (FA1) (SoSe 2016)

News (21.07.2016): If you are a student in Physics, and you passed the exam, your mark (Note) will not be reported to the Physics department automatically. You need to download a 'Schein', fill it out, and bring it to the secretariat. (Once it has later been signed, you may pick it up the same place).

Content of the lecture (Kurzübersicht der Vorlesung)

Please sign up for the Exercises. (Bitte zu den Übungen anmelden.)

Time and place (Zeit und Ort):

Lecture (in English!) (Vorlesung) (Prof. Sørensen): Tue 12.30-14 & Thu 8.25-9.55 in C 123.

Homework discussion (Zentralübungen) (A. Groh): Mo 16-18 in C 123. (First session: Mo 18 April.)

Tutorials/Tutorien: See separate webpage for time and place. (First session: Tue 19 April.)

Synopsis (Kurzbeschreibung):
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 (PDE's), 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 and the spectral theorem for compact operators. These are powerful tools for applications to PDE's and quantum mechanics, respectively. (More details on content below.)

NB Die Vorlesung wird auf Englisch gehalten.

Audience (Hörerkreis): Students of Mathematics, Wirtschaftsmathematik, and Physics.
("Gilt für Bachelorprüfungen Mathematik (WP4) und Wirtschaftsmathematik (P12), Masterprüfung Wirtschaftsmathematik (WP11), Diplomhauptprüfung Mathematik (RM,AM), Diplomhauptprüfung Wirtschaftsmathematik (Kernfach D)").

Prerequisites (Vorkenntnisse): 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/Klausur (on Saturday July 16th, 2016 (in the morning)). 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 exam/Klausur is raised by 0.3/0.4 EXCEPT for marks below 4.0 (and, for 1.0 of course!)

Exam (Prüfung): There will be a written exam on Saturday July 16th, 2016 (in the morning). More information is here.
(Es wird eine Klausur geben, am Samstag 16. Juli 2016 (vormittags); mehr Information (in Englischer Sprache) befindet sich hier.)

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 Groh and Prof. Sørensen (and all tutors - except Francesco Romano) 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 exam/Klausur will be in English, but the solutions can be turned in either in German or in English.

Literature (Literatur): There will be no comprehensive Skript (no 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 content of the lecture will keep you updated. Contents (preliminary - not necessarily in this order):
  • 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; examples: Lp-spaces, Hölder spaces, Sobolev spaces; linear operators; linear functionals and dual space; Hilbert spaces).
  • 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).
Homework discussion and Tutorials (Ablauf des Übungsbetriebs):
For all information, see the separate webpage.

Office hours (Sprechstunden):
Prof. Sørensen: Thu 10-11 (Room/Raum B 408) or by appointment via email (oder nach Vereinbarung via Email).
A. Groh: Mo 14-15 (Room/Raum B 410).
S. Gottwald: Fr 11-12 (Room/Raum B 403).


Letzte Änderung: 26 August 2016 (no more updates).

Thomas Østergaard Sørensen

Curriculum Vitae