Lecture Course: Functional Analysis

Tue, Thu 14 – 16   in B 006

Organisation of tutorials: Martin Gebert and Heinrich Küttler

Tutorials, problem sheets and further information

News

• 18/7/13   Second exam (Nachholklausur): Tuesday, 8 October 2013, 12:00 – 15:00 in B 051 and B 052.
• 5/7/13   Typo in Theorem 3.43 corrected: inequality in the claim was reversed.
• 3/7/13   More details given in the proof of Thm. 4.28(d).
• 3/7/13   Proof of Thm. 4.32: The constant K was redefined which avoids a circularity (reference to Lemma 4.31 dropped out).
• 18/6/13   A password protected list with the scores in the midterm exam is available now. Please note that the information given there is inofficial. Official scores are only available in the review and inspection session on Wed 19 June 2013 from 4:00 – 6:00 pm in B 052. You find the inofficial list here. The password is given in the lecture and will not be communicated by email.
• 15/6/13   Proof of Theorem 3.50: the argument leading to the equality of ξ and f*g has been expanded in the script.
• 15/6/13   Theorem 3.47: regularity of the measure μ was forgotten as a hypothesis of the theorem in the lecture. See the script for the correct formulation.
• 5/6/13   Definition 2.47 on orthogonal complements extended. Originally only formulated for subspaces, now for arbitrary subsets.
• 8/5/13   Proof of Theorem 1.34 corrected (allow for base elements not fully contained in a set of the cover)
  and formulation of Theorem 1.50 improved (only pointwise boundedness used in the proof).

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

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]