Functional Analysis - Summer term 2012 (SoSe 2012)
Results Make-up exam / Wiederholungsklausur:
You can have a look at your graded test (`Klausureinsicht') and pick up the Schein at
B-408 (Prof. Sørensen), on
MONDAY October 22 at 12:00 -- 13.00 or TUESDAY October 23 at 12:00 -- 13.00.
The original test stays with us but you can have a look at it.
If, for some reason, you do not agree with the marking, you may hand
in your written complaint including justification when you hand back
your test.
Please
pick up uncollected, marked homework the same time and
place (last chance!).
You are not allowed to see the exam or pick up
the Schein for someone else (bring ID / Ausweis!).
If it is _completely_ impossible for
you
to come Mon/Tue you may come to the secretary (in B-411) _after_
October 24th.
Note: The bonus for the
homework was counted also for the Make-up exam /
Wiederholungsklausur.
The exam (including
solutions; with no warranty of elegance nor uniqueness). Removed.
Extra sessions (exam preparation):
We will discuss a selection of the extra exercises.
Time and place:
Monday 15. October 2012, 18:15-20:00 in B-138
Tuesday 16. October 2012, 18:15-20:00 in B-005
Exam results:
You can have a look at your graded test (`Klausureinsicht') and pick up the Schein (see below) at
B-408 (Prof. Sørensen), on
MONDAY July 23 at 15:00 -- 17.00 or TUESDAY July 24 at 10:00 -- 11.00.
The original test stays with us but you can have a look at it.
If, for some reason, you do not agree with the marking, you may hand
in your written complaint including justification when you hand back
your test.
You can
pick up uncollected, marked homework the same time and
place.
You are not allowed to see the exam or pick up
the Schein for someone else (ID / Ausweis!).
If it is _completely_ impossible for
you
to come Mon/Tue you may come to the secretary (in B-411) _after_
August 5th.
Make-up exam / Wiederholungsklausur:
Time and place: Saturday 20. October 2012,
09:00-11:00 (in B 052 and ??).
Material and conditions: As for the Exam/Klausur on July 16th
(see below).
If you wish to participate, sign up on
the `Lecture
Assistant'
(under `Funktionalanalysis (Wiederholungsklausuranmeldung) (Soerensen)', group `WI-KlausurOkt20')
before
Wednesday Aug 29 12:00 (noon) Monday Oct
1st 12:00 (noon) (EXTENDED!). Please fill
in everything including email address.)
NOTE: Change of rules of participation!:
EVERYONE who participated
in the Exam/Klausur (on July 16th) (_and_ signed up as
described above) may participate in the
Make-up exam/Wiederholungsklausur (even _if_ they passed the Exam on
July 16th).
(Jeder, der/die an dem Klausur am 16 Juli teilgenommen hat, darf,
nach Anmeldung, am Wiederholungsklausur am 20 Okt teilnehmen.)
If you _failed_ the exam on July 16th, you may participate in the
mark-up exam/Wiederholungsklausur on Oct 20th (if and only if you sign
up as described above).
If you _passed_ the exam on July 16th, and do NOT sign up as
described above (at the latest by August 29th), your mark (`Note') from July 16th will be reported to the
Pruefungsamt (on Aug 29th). After this, the mark _cannot_ be
changed. - After this, you may pick up
your Schein at the secretary
(in B-411).
If you _passed_ the exam on July 16th, and _do_ participate in the
Make-up exam/Wiederholungsklausur, your final mark will be the best of
the two marks, from July 16th and that of Oct 20th.
Preparation (suggestions):
Read and understand your lecture notes.
Go through the Exercises and Problems from class again (their solutions are
online until the mark-up exam/Wiederholungsklausur Oct. 20th).
Don't only do the
details, also try to understand, in each exercise, what was the main idea that did the
trick.
Some extra, exam-relevant, exercises to practise on (zum ueben) are
here
(solutions will _not_ be given). Also, there will be one or more extra exercise
sessions (on _some_ of these exercises, plus questions) in October
(see above).
(For all questions, please email Prof. Sørensen - subject `Functional Analysis'.)
++++++++++++++++++++++++++++++++++++++
Last week of class (July 16 - 20):
Lectures: As usual.
Exercise session: As usual (Discussion of exam).
Tutorium/repetitorium
sessions: NONE !!
Endklausur (Final (written) Exam):
Time: Monday 16 July 18:30-20:30.
Place: Room B-052.
See all information below.
Time and place:
Lectures (Prof. Sørensen):
Monday 14:15--16:00, Room B 005.
Wednesday 16:15-18:00, Room B 005.
Exercise
sessions ("Übungen") (Alessandro Michelangeli):
Tuesday 18:15--20:00, B 005.
Tutorium/repetitorium
sessions (Zhigun, Llosa, Staffler, Michelangeli, Sørensen):
Tuesday 10:15--12:00, Room B 045. (T. Sørensen - Starts April 24th.)
Wednesday 10:15--12:00, Room B 039. (A. Michelangeli - Starts April 25th.)
Wednesday 14:15--16:00, Room B 045. (C. Llosa - Starts April 25th.)
Thursday 10:15--12:00, Room B 045. (A. Zhigun - Starts April 26th.)
Thursday 16:15--18:00, Room B 133. (B. Staffler - Starts April 26th.)
(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, 10:15--11:00.
Alessandro Michelangeli (Room B 334): Tuesday, 17:15-18:00.
Claudio Llosa (Room B 045): Wednesday 16:00 (after tutorial).
Anna Zhigun (Room B 045): Thursday 12:00 (after tutorial).
Benedikt Staffler (Room B 133): Thursday 18:00 (after tutorial).
Graders/Korrektoren:
See the webpage of
the Exercises.
General information for the course:
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: to be announced; more details later).
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: Monday 16 July 18:30-20:30
Place: Room B-052
Material: Complete material of all lectures up until
and including
Wednesday July 11th, and Exercise sheets up until and including
Sheet 12.
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.
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). If necessary,
there will be another
exam (Wiederholungsklausur) offered
later for those who
failed the first one (Saturday October
20th, 09:00-11:00). 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 to improve the grade on the first one
(III. Paragraph 11 (7) of the Prüfungsordnung).
The second exam will be similar to the first one, 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 first exam for a properly
documented health reason (see III. Paragraph 11. (5) Satz 4--7 of the Prüfungsordnung).
(In particular, one has to participate in the first Klausur to be allowed to
participate in the second one.)
Exercise Sheets:
Will be posted on the web every Tuesday by 20:30 (8:30pm) on the
exercise homepage.
Solutions are due the
following Tuesday at 18:00 (6:00pm) (sharp!) in the designated box. First sheet is posted
on April 17th. 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, Dr. Michelangeli and Prof. Sørensen (and all tutors) 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 (maybe!) find the more precise
references.
- M Dobrowolski, Angewandte Funktionalanalysis, Springer,
2006. (Electronic access - from LMU only).
- W Kaballo, Grundkurs Funktionalanalysis, Spektrum
Akademischer Verlag, 2011.
- D Werner, Einführung in die Funktionalanalysis,
Springer, 2007.
- M Reed and B Simon, Methods of modern Mathematical Physics I:
Functional analysis, Academic Press, 1980.
- P D Lax, Functional Analysis, Wiley, 2002.
- Rudin, Real and Complex Analysis, McGraw and Hill, NY,
1987.
- Rudin, Functional Analysis, McGraw and Hill, NY, 1991.
- Marcel Griesemer, Real Analysis (A survey of integration theory), UAB 1999-2000.
-
W A Sutherland, Introduction to metric and topological spaces,
Oxford University Press, 2009.
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; 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:
-
Euclid,
Hausdorff,
Cauchy,
Banach,
Riesz,
Schwarz,
Hoelder,
Zorn,
Hilbert,
Fourier,
Bessel,
Parseval,
Ascoli,
Arzela,
Lebesgue,
Minkowski,
Fischer,
Radon,
Nikodym,
Beppo-Levi,
Sobolev,
Meyers,
Serrin,
Lipschitz,
Rellich,
Kondrashov,
Baire,
Steinhaus,
Hahn,
Alaoglu,
Last update: October 21st, 2012 by Thomas Østergaard Sørensen.