Vorlesung: Functional Analysis (SoSe 2025)
Lecture (Vorlesung):
Tue 12-14 & Thu 12-14 (both in B 005). LSF
Exercises (Übungen):
See separate webpage (Moodle). LSF
Tutorials (Tutorien):
See separate webpage (Moodle).
Synopsis (Kurzbeschreibung):
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 (PDE's), calculus of variations, 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 and the spectral theorem for compact operators. These are powerful tools for applications to PDE's and quantum mechanics, respectively. (More details on content below.)
NB Die Vorlesung wird auf Englisch gehalten.
Audience (Hörerkreis):
Students pursuing the following degrees: BSc Mathematics, BSc Financial Mathematics.
Credits:
9 (6+3) ECTS.
Prerequisites (Vorkenntnisse):
Analysis I-III, Lineare Algebra I-II. You find handouts with the needed facts (without proofs, and to be updated!) below, and in Moodle.
Language (Sprache):
English.
Exam (Prüfung):
See Moodle.
Content (Inhalt):
- Recapitulation of basic notions PDF (Handout: Topological and metric spaces (Ana2))
0.1 Topological spaces
0.2 Metric spaces - Topological and metric spaces
1.1 Limits and continuity PDF
1.2 Metric spaces PDF
1.3 Example: sequence spaces l^p PDF
1.4 Compactness
1.5 Example: spaces of continuous functions
1.6 Baire's theorem
- Banach and Hilbert spaces
2.1 Vector spaces
2.2 Banach spaces
2.3 Linear operators
2.4 Linear functionals and dual spaces
2.5 Hilbert spaces - L^p-spaces (Handout: Measure and Integration Theory (Ana3))
3.1 Completeness and dual space
3.2 Separability - The cornerstones of functional analysis
4.1 Hahn-Banach theorem
4.2 Three consequences of Baire's theorem
4.3 (Bi-) Dual spaces and weak topologies - Bounded operators
5.1 Topologies on the space of bounded linear operators
5.2 Adjoint operators
5.3 The spectrum
5.4 Compact operators
5.5 Fredholm alternative and the spectral theorem for compact operators
In Moodle you will find a copy of the notes from the lecture (to be updated as we go along).
Above you will find a short description of the content of the lecture.
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!). Most of them cover the material of a two-semester course.
Supplementary literatur (Ergänzende Literatur):
- H. W. Alt, Lineare Funktionalanalysis, 6. Auflage, Springer, 2012.
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.
- C. Clason, Einführung in die Funktionalanalysis, 2. Auflage, Birkhäuser, Cham, 2025.
- M. Dobrowolski, Angewandte Funktionalanalysis, 2. Auflage, Springer, 2010.
- F. Hirzebruch and W. Scharlau, Einführung in die Funktionalanalysis, Spektrum Akademischer Verlag, 1996.
- W. Kaballo, Grundkurs Funktionalanalysis, 2. Auflage, Springer Spektrum, 2018.
- E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Classics Library, 1978.
- P. D. Lax, Functional Analysis, Wiley, 2002.
- M. Reed and B. Simon, Methods of modern mathematical physics, Volume I: Functional analysis, Academic Press, 1980.
- G. Teschl, Topics in Linear and Nonlinear Functional Analysis, Graduate Studies in Mathematic, AMS, to appear.
- D. Werner, Funktionalanalysis, 8. Auflage, Springer Spektrum, 2018.
Office hours (Sprechstunde):
See Moodle.
To access the course material, you need to sign up (closes 07. May 2025) in Moodle here (Psword: Banach) .
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Letzte Änderung: 30 April 2025.
Thomas Østergaard Sørensen