Functional Analysis  Summer term 2011 (SoSe 2011)
Makeup exam / Wiederholungsklausur:
Friday 21. October 2011,
16:0018:00 (in B005 and B006). Conditions: See below.
If you wish to participate, please send an email to either Oliver
Matte or Thomas Sorensen.
(Wiederholungs)Klausureinsicht: Tue Oct 25th and Thu Oct 27, 13:1514:00 in B408 (prof Sorensen).
Time and place:
Lectures (Prof. Sørensen):
Wednesday/Friday 8:1510:00, Room B 051. (Starts May 4th).
Exercise
sessions ("Übungen") (PD Oliver Matte):
Monday 16:1518.00, B
051. (Starts May 9th.)
Change last week:
Lectures: Monday July 25th, 16:1518.00, Wednesday July 27th, 8:1510:00. Exercise
sessions Friday July 29th, 8:1510:00 ("Klausurvorrechnung").
Tutorium/repetitorium
sessions (Torben Krüger, Matte, Sørensen):
Tuesday 08:1510:00, Room B 004. (Oliver Matte  Starts May 10th.)
Wednesday 10:1512:00, Room B 040. (Thomas Sørensen  Starts May 11th.)
Thursday 08:1510:00, Room B 040. (Torben Krüger  Starts May 12th.)
Thursday 16:1518:00, Room B 047. (Torben Krüger  Starts May 12th.)
(For organisation and programme for the exercises, and tutorium, see
the link above.
To participate, you need to sign up
here.)
Office hours:
Thomas Østergaard Sørensen (Room B 408): Thursday, 10:0011:00
Oliver Matte (Room B 404): Monday, 14:0015:00
Torben Krüger (Room B 208): Thursday, 10:0011:00
Graders/Korrektoren:
See the webpage of
the Exercises
(PD Matte).
General information for the course:
Synopsis:
Functional analysis can be viewed as ``linear algebra on
infinitedimensional 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, HahnBanach 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 IIII, Lineare Algebra III).
Requirement for passing the course:
The course ends with a written final exam (time: Friday, July 22nd,
15:0017:30; 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):
Time: Friday, July 22nd, 15:0017:30.
Place: Schellingstr. 3 (!), S 001.
Material: Complete material of all
lectures up until and including
Friday July 15th, and Exercise sheets up until
and including Sheet
10. The solutions to Sheet 10 will be
posted on July 18.
You are allowed to bring 1 (one) twosided cheat sheet (Spickzettel):
a piece of A4paper on which you have written (on both sides) whatever
you think might be useful for the Klausur/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.
The final grade is determined by the final exam plus the bonus points
from homework, see the precise rules above.
There will be no makeup exam (keine Nachholklausur). If necessary,
there will be another exam (Wiederholungsklausur) offered around mid
October for those who
failed the first one (precise date is to be fixed). This will be a
new exam, i.e. all grades from the first exam will be reported to the
Prüfungsamt. According to the Exam Rules (Prüfungsordnung), the
second exam cannot be used the improve the grade on the first one
(III. Paragraph 11 (7) of the Prüfungsordnung).
The exam in October will be similar to the one on July 22nd, but bonus
points from the homework will not be counted; the grade will be
determined solely by the result on that exam. This restriction rule
will not apply to those who missed the exam on July 22nd for a properly
documented health reason (see III. Paragraph 11. (5) Satz 47 of the Prüfungsordnung).
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 nonGerman 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 twosemester 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 Lpspaces.
(Measures; integration; Lpspaces; decomposition of measures).

The cornerstones of functional analysis.
(HahnBanach 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,
BeppoLevi,
Hahn,
Steinhaus,
Alaoglu,
Last update: October 11th, 2011 by Thomas Østergaard Sørensen.