Lecture Course: Functional Analysis

Tue, Thu 10 – 12   in B 051

Organisation of tutorials: Sebastian Gottwald and Andreas Groh

Tutorials, problem sheets and further information

News

• 3/4/14   The central tutorial class will take place on Wednesdays from 12 – 14 in B006 (change!).
  On Wednesday 9/4/14 and 16/4/14 there will a lecture instead of a tutorial 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

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]