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Lecture Course: Functional Analysis

Tue 12 – 14, Thu 10 – 12   in B 006


Organisation of tutorials: Florian Behr and Marcel Schaub

Tutorials, problem sheets and further information

News

  • 9/10/19   The results of the make-up exam of 9/10/19 are now available (only of those participants who agreed to it). Access requires the same login data as for the script. You can inspect the corrected exam sheets (Klausureinsicht) on Friday, 11 October 2019 between 10:00 and 11:00 in B439.
  • 21/8/19   Participation in the make-up exam on 9/10/19 requires registration in the UniWorX system no later than Friday, 27 September (23:59).
  • 17/8/19   The make-up exam will take place on Wednesday, 9/10/19 at 10:00 in B139. Participation is by registration only. Details will be given on 21/8/19.
  • 17/8/19   The results of the exam of 29/7/19 are now available. Access requires the same login data as for the script. You can inspect the corrected exam sheets (Klausureinsicht) on Tuesday, 20 August 2019 between 14:00 and 15:30 in B349.
  • 21/7/19   Last year's exam is available here (use the same login data as for the script). I also corrected several misprints in the script; thanks to Johannes Bartenschlager for pointing them out to me.
  • 12/7/19   Participation in the exam requires registration no later than Tuesday, 23 July (23:59). See here for more details.
  • 28/6/19   The presentation of the Hahn-Banach theorem has changed in the script. The former Theorems 4.3 and 4.4 are now combined into one theorem.
  • 25/6/19   The exam will take place on Monday, 29 July 2019 at 10:00. Details will follow.
  • 18/6/19   Added a (hopefully clarifying) comment in the script on the use of representatives in Lemma 3.15.
  • 13/6/19   The definition of Lpc from the lecture has been corrected in the script.
  • 1/5/19   Problem sheet 2 is due on 8/5/19 at 14:00. And here is Tutorial sheet 2.
  • 30/4/19   Tutorial sheet 1.
  • 24/4/19   Problem sheet 1 is due on 2/5/19 at 14:00.
  • 24/4/19   Please register for the exercise and tutorial classes. Deadline for registration: Friday, 26/4/19 at 16:00.
  • 24/4/19   On Monday, 29/4/19 there will be a lecture at 16:15 in B 006 instead of an exercise class.

Synposis
Functional analysis can be viewed as "linear algebra on infinite-dimensional vector spaces". 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: topological foundations, Banach and Hilbert spaces, dual spaces, Hahn-Banach thm., Baire thm., open mapping thm., closed graph thm., weak topologies. If time permits we will also cover Fredholm theory for compact operators and the spectral theorem.

Prerequisites
Analysis I – III, Linear Algebra I, II
The following basic notions about topological and metric spaces will be used throughout from lecture one.

Audience
Students pursuing the following degrees: BSc Mathematics, BSc Financial Mathematics, MSc Financial Mathematics

Literature
The course will not follow a particular textbook. The following list 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.
  • M Reed and B Simon, Methods of modern Mathematical Physics I: Functional analysis, Academic Press, 1980
    [excellent textbook with a focus on spectral theory, beginning not very gentle, proofs sometimes a bit brief; unfortunately rather pricey]
  • D Werner, Einführung in die Funktionalanalysis, Springer, 2007
    [a German classic, covers a broad range of topics, including historical remarks]
    Full text with LMU library account
  • M Dobrowolski, Angewandte Funktionalanalysis, Springer, 2006
    [the basics of functional analysis plus a thorough discussion of Sobolev spaces and elliptic PDE's]
    Full text with LMU library account
  • E Kreyszig, Introductory functional analysis with applications, Wiley, 1978
    [thorough and pedagogical, very explicit proofs, does not cover all topics treated in the course (e.g. no Lp-spaces)]
  • P D Lax, Functional Analysis, Wiley, 2002
    [well readable with an emphasis on spectral theory and some applications to quantum mechanics]
  • F Hirzebruch and W Scharlau, Einführung in die Funktionalanalysis, BI Mannheim, 1971
    [another German classic, elegant but very(!) concise]

Script
Version of 19 August 2019