TU Chemnitz: Fakultät für Mathematik: Professur Analysis
Functional analysis
Vorlesung in englischer Sprache im Wintersemester 2002/2003 an der TU Chemnitz. Informationen im Vorlesungsverzeichnis,
Coordinates:
Lectures and exercise classes: Do 1. 2/N 001, Fr 2. 2/B102
Introduction:
Functional analysis is a field that has a central position between applied and
pure fields that deal in one way or the
other with function spaces.
Of course, a thorough understanding of basic analysis and linear algebra is necessary
to follow the course. There is no
text book for the lecture but literature for complementary reading will be posted below.
The exercises
will consist of two different components: Exercise sheets that students should work on on their own and working
sections that will be embedded in the regular course:
Exercise sheets:
Sheet 1: 
Solutions
Sheet 2:
Solutions
Sheet 3:
Solutions
Sheet 4:
Solutions
Sheet 5:
Solutions
Sheet 6:
Working sections:
Section A: Nets and convergence
Section B: Finite vs infinite dimensions
Section C: subspaces, quotients and their duals
Section D: completion of normed, metric and pre-Hilbert spaces
Questions that might come in the oral exam:
Content:
Introduction
0. Banach spaces: the first encounter
1. Basic structures
1.1 Linear algebra
1.2 Metric spaces and topology
Working section A: nets and convergence
1.3 Norms and scalar products
1.4 Linear operators
Working section B: Finite vs infinite dimensions
Working section B: solution proposed by K. Luther and D. Oriwol
2. The Hahn-Banach theorem
2.1 The Hahn-Banach extension theorem
2.2 The Hahn-Banach separation theorem
2.3 The bipolar theorem
Working section C: subspaces,
quotients and their duals
3. Baire's theorem and its consequences
3.1 Baire's theorem
3.2 The open mapping theorem and Banach's isomorphism
theorem
3.3 The closed graph theorem
3.4 The uniform boundedness principle
3.5 The Banach-Steinhaus theorem
4. Dual spaces and adjoints
4.1 Examples of dual spaces
4.2 Adjoint operators and the closed range theorem
Working section D: completion of normed, metric and pre-Hilbert spaces
5. More spaces
5.1 Hilbert spaces: basic geometry
5.2 Spaces of continuous functions
To be continued ...
Any questions? The answer is ... here
Literature:
- Functional Analysis
K. Yosida
Springer, Berlin 1968
- Funktionalanalysis: Ein Arbeitsbuch
M. Mathieu
Spektrum Akademischer Verlag, Heidelberg 1998
- Lineare Funktionalanalysis: Eine anwendungsorientierte Einführung
H.W. Alt
Springer, Berlin 1992
- Einführung in die Funktionalanalysis
R. Meise, D. Voigt
Vieweg, Braunschweig 1992
