Vorlesung: Functional Analysis (SoSe 2022)

[UPDATE 18.04.2022:] This course will be entirely live / face-to-face ("Präsenz") (lectures, exercise classes, tutorials).
Please note the present LMU Corona-rules.

To access the course material, you need to sign up in uni2work here.

Lecture (Vorlesung):
Tue 14-16 & Thu 10-12 (both in B 006).   LSF

Exercises (Ãœbungen):
See separate webpage (uni2work).   LSF

Tutorials (Tutorien):
See separate webpage (uni2work).

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 uni2work.

Language (Sprache):
English.

Exam (Prüfung):
See uni2work.

Content (Inhalt):

0. Recapitulation of basic notions   (Handout: Topological and metric spaces (Ana2))

   0.1 Topological spaces
   0.2 Metric spaces

1. Topological and metric spaces

   1.1 Limits and continuity
   1.2 Metric spaces
   1.3 Example: sequence spaces l^p
   1.4 Compactness
   1.5 Example: spaces of continuous functions
   1.6 Baire's theorem
   

2. 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

3. L^p-spaces   (Handout: Measure and Integration Theory (Ana3))

   3.1 Completeness and dual space
   3.2 Separability

4. 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

5. 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

Literature:
In uni2work 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, Birkhäuser, Cham, 2019.
• 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.

See also this discussion.

Office hours (Sprechstunde):
See uni2work.

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Letzte Änderung: 03 August 2022 (No more updates).

Thomas Østergaard Sørensen