Lecture Course: Functional Analysis

Mon, Wed 12 – 14   in B 005

Organisation of tutorials: Sabine Bögli

Tutorials, problem sheets and further information

News

• 10/10/18   We have revised the grading of the make-up exam. The revision does not diminish the number of points of anybody but leads to an increase in points for some of the participants. The updated results can be found here.
• 9/10/18   The results of the make-up exam of 9/10/18 are now available. Access requires the same login data as for the script. You can view the corrected exam sheets (Klausureinsicht) on Monday, 15 October between 15:00 and 16:00 in B439.
• 25/7/18   The make-up exam will take place on Tuesday 9/10/18 at 9:00 in B005. Participation requires registration by email to me no later than 1/9/18. Use the subject 'make-up exam FA' and include your name, degree course (Studiengang) and matriculation number (this is independent of the previous course registration via Lecture Assistant).
• 25/7/18   The results of the exam of 24/7/18 are now available. Access requires the same login data as for the script. You can view the corrected exam sheets (Klausureinsicht) on Thursday, 26 July between 15:00 and 16:00 in B335 (Office Sabine Bögli).
• 11/7/18   The complete script is now online. Changes: (i) streamlined the proof of the analytic Fredholm theorem (γ_n(z) eliminated), (ii) introduced a new Remark 5.32, (iii) the first part of the proof of Theorem 5.29 (formerly Theorem 5.28) is now a statement on its own, the new Lemma 5.9.
• 4/7/18   Participants are allowed to use a cheat slip in the exam: size at most A4, front and back side, only handwritten!
• 27/6/18   The proofs of Lemma 4.25 (b) and (d) in the previous version of the script were put in a more abstract setting and are now included as a new Lemma 4.24. This allows for a straightforward application of the statements in the context of the weak* and the strong and weak operator topologies. Note that the numbering in Section 4 has shifted by one from 4.24 on as a result of this new lemma.
• 21/6/18   Remark 4.23(d) was added in the script, as well as the condition that X is a topological vector space in Lemma 4.25(b).
• 12/6/18   The written final exam will take place on Tuesday, 24 July at 2 pm in C123. It requires registration by email to Sabine Bögli no later than 6 July. Use the subject title 'Exam FA' and include name, degree course (Studiengang) and matriculation number (this is independent of the course registration via Lecture Assistant).
• 25/4/18   The lecture on Wednesday, 9 May 2018 will not take place. Instead, there will be an additional lecture on Thursday, 17 May 2018 at 14:15 in B 138.
• 25/4/18   The lecture on Wednesday, 2 May 2018 will start at 12:30 and end at 14:00.

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]
• M Dobrowolski, Angewandte Funktionalanalysis, Springer, 2006
  [the basics of functional analysis plus a thorough discussion of Sobolev spaces and elliptic PDE's]
• 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 L^p-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]