Functional Analysis 2 - Winter term 2011/12 (WiSe 2011/12)
Thomas Østergaard Sørensen
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).
Exercise
sessions ("Übungen") (A. Michelangeli):
Wednesday 18:15--20.00, Room C 111. (Starts Oct 25th.)
Sign up here.
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.
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,
Last update:
February 7th
, 2012 by Thomas Østergaard Sørensen.