# 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

Tutorials, problem sheets and further information

**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]