Functional Analysis (in English)
Home page for the course (WS2004)
(Prof. Laszlo Erdos and Dr. Thomas Sorensen)
FINAL EXAM
Out of 40 points, the following results were obtained in
decreasing order:
40, 33, 29, 29, 29, 26, 24, 24, 21, 18, 17, 17, 16, 13, 8, 2
Exams are available at Frau Winter's office (Room 117)
between 9:00-12:00. You can have a look at the exams in Frau
Hoechst office (Room 330 between 8:30-12:30), but we keep
the originals.
The Scheins will be available (Frau Hoechst office) approximately
around Feb 16, after the last homework set is graded.
SOLUTIONS
Solutions
RULES
Time, place: February 5, Saturday, 15:30-17:30. Room E06.
The final is obligatory for the Schein (see rules below).
If you have an absolute unaviodable conflict, let me know
as soon as possible.
Material: Complete material of all lectures
up to the lecture on Jan 28 (inclusive).
Correspondingly, every Exercise sheet up to Sheet no. 12 (inclusive).
The solutions of Sheet 12 will be published on Feb 4. at 2:00 PM
(after the return deadline).
You are allowed to bring 1 (one) two-sided cheat
sheet: a piece of A4-paper 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.
Put your name on every sheet you wish to hand in, and write readable.
There will be extra office hours:
Friday, Feb 4. 15:00-16:00, Thomas Sorensen (office 335)
Saturday, Feb 5, 13:00-15:00, Laszlo Erdos (office 329)
General Information for the course
For:
Students in the International Master Program, Students of mathematics and physics.
Prerequisites:
Introductory courses in analysis and linear algebra
(Analysis I-III, Lineare Algebra I-II)
Certificate (Schein):
Gilt für Diplomhauptprüfung (AM); Hauptprüfung für das Lehramt an
Gymnasien.
Requirement for the Schein:
Due the recent budget cuts, we do not know yet how
many homework problems we can afford to grade, so the following
rule might change until the end of the first week.
There will be weekly exercise sheets, but we probably
will not be able to grade all of them. Most likely
we can grade only half of the problems from each
sheet. In this case, the problems to be graded will
be determined randomly.
Exercise sheets and the Final exam
both count with equal weight (50%-50%) towards the total performance.
To obtain the Schein, 50% total performance and
at least 40% performance in the Exercise Sheets
and at least 40% performance in the Klausur are required.
Time and place:
Lectures (Prof. Erdos):
Tuesdays and Fridays, 9:15-11:00, Room E5
Starts: October 19.
Exercise sessions (Dr. Sorensen):
Tuesday 14:15-16:00, Room E47,
Wednesday 16:15-18:00, Room E47.
Office hours:
Prof. Erdos: Thursdays 2-4. Room 329 (block B).
Dr. Sorensen: Thursdays 1-3. Room 335 (block B).
Klausur (Final Exam):
Feb 5 (Saturday) at 15:30.
Room and more details will follow.
Exercise Sheets:
Posted on the web every Friday.
Solutions are due the following Friday at 2pm in the
designated box.
What is functional analysis?
It is a deep fact of the physical world around us that most of its behavior can be formulated in
terms of differential and integral calculus. Wave and heat propagation, elasticity, motion of
galaxies and electrons etc. are all described by (partial) differential equations (PDE). Functional
analysis is the starting point for mathematical analysis in real-life physical systems, in particular
it is the first step towards PDE's and numerical methods. It is the child of two fundamental
branches of mathematics: analysis and linear algebra. In analysis we have learned how to grasp
infinite procedures (e.g. limits) rigorously, while linear algebra has taught us how to deal with
finitely many (linearly) interrelated scalar quantities in a computationally effective way. A water
wave or an elastic sheet, however, is described by a continuum of interrelated scalars (think of
the displacement of each point in the wave), so one must understand how to do linear algebra in
infinite dimensions. Therefore the powerful concept of the limit from analysis became
indispensable and functional analysis was born. As a prodigy child, very quickly after its birth, it
has proved to be much more far-reaching than a refined synthesis of known mathematical ideas.
In the late 20's it turned out that the foundations of quantum physics rely entirely on functional
analysis. It has also revolutionized the theory of PDE's by providing solid ground for the theory
of distributions, which made it possible to solve a much wider class of PDE's. This course will
present the standard introductory material to functional analysis with more focus on
applications. The two fundamental results are the Fredholm theory of compact operators that
enables us to solve simple PDE's and the spectral theorem which is the cornerstone of the
mathematical model of quantum mechanics.
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. Sorensen is 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. Some
additional web-notes will be published on
particular topics.
The brief contents of the lectures
will keep you updated, here you will find the precise references.
Our main text
Reed-Simon: Functional Analysis (Methods of Modern Mathematical Physics Vol. I), Academic
Press, 1980.
1 copy of this book is kept on reserve in the library.
The book is also available for purchase
(see, e.g., www.amazon.de), but it is quite pricey
(although it's worth its price). Buying the book is not required.
Additional texts:
- Lieb-Loss: Analysis. Amer. Math. Soc. 2001.
- Peter Lax: Functional Analysis. Wiley, 2002
- Werner: Funktionalanalysis (in German). Springer 2000.
- Rudin: Real and Complex Analysis. McGraw and Hill, NY, 1987
- Rudin: Functional Analysis. McGraw and Hill, NY, 1991.
-
Liskevich: Measure Theory. Webnotes . Elemetary and detailed
description of the Lebesgue measure.
-
Wahl: Funktionalanalysis (In German) PDE-oriented discussion of
functional analysis, many explicit calculations with function spaces.
- Scherer: Funktionalanalysis (In German)
A lot of material in a very dense form. Good reference. Includes a complete introduction
to Lebesgue measures and integration.
- Ward: Functional Analysis
Elementary discussion with many details. Includes good explanation of Fourier transform.
-
Chelminsky Funktionalanalysis (in German). Lot of extra material:
fixpoint theorems, distributions, Sobolev spaces, semigroups.
Contents: (preliminary)
- Review of linear algebra
- Review of basic concepts of analysis.
- Metric and normed spaces.
- Review of the Lebesgue integral and measure.
- Hilbert spaces. Fourier transform.
- Operators on Banach spaces
- Duals and weak convergence.
- Hahn-Banach and Riesz-Markov theorems.
- Baire category theorem and its consequences: Banach-Steinhaus,
Open Mapping and Closed Graph theorems. Weierstrass approximation theorem.
- Quick introduction to topology. Compactness. Banach-Alaouglu theorem.
- Topologies of bounded operators. Adjoints.
- Spectrum
- Compact operators. Fredholm alternative for Hilbert spaces.
Singular value decomposition for compact operators. Solution to the
Dirichlet problem. Polar factorization. Trace.
- Spectral theorem for bounded self-adjoint operators. Functional calculus.
Spectral projections, spectral types.
- Outlook. Applications in quantum mechanics.