Vorlesung „Functional analysis“

• Target group: students of mathematics, physics, and computer science in semester 4 and higher
• Lecturer: Thomas Schick
• Subject: funktional analysis
• Area: analysis
• Prerequisits: Diff I/II, AGLA I/II
• Time: Mo, Th, 12:15–13:55
• Place: Maximum
• contact/questions: thomas.schick@math.uni-goettingen.de, Tel. 39-27766
• exam: written after the teaching period; 50% solved exercises

Funktional analysis (FA) is a of fundamental importance for the analytical branch of pure mathematics, and in the much of applied and numerical mathematics, e.g. in the theory and numerics of partial differntial equations and integral equations, for optimization and in approximation theory, and also in many branches of stochastics.

At the same time, it plays an important role in physics, in particular in quantum mechanics. Many parts of funktional analysishave been developped at the beginning of the twentith century, to lay solid mathematical foundations for the ideas of the physicists. As a principle: quanten mechanical systems are described by a Hilbert space, measurments by application of a self-adjoint operator, the time evolution via application of an operator which has to be inserted in a function.

FA is (after Diff I and Diff II) the completion of the analytical basic education of any mathematician.

The subject of the course FA is the theory of normed vector spaces, in particular of Hilbert spaces, and the continuous linear maps between these.

We will cover in particular

• Hilbert spaces and their geometry
• Banach spaces, their generalizations and the basic principles of funktional analysis (Hahn-Banach, open mapping, …theorems)
• bounded operators and their pectrum
• spektral theorem for (self adjoint) operators on a Hilbert space: substituing operators in funktions
• compact operators and their spektral theory: solution theory of linear equations (Riesz-Fredholm theory)
• probably: distributions (=generalized functions)

The course is planned to be held in english, on mutual agreement we could also switch to German (or French or Italian)

Literatur:

• T. Schick: short lecture notes (to be adapted during the course)
• W. Rudin: Functional Analysis
• M. Reed, B. Simon: Functional Analysis I, II
• H. Schröder: Funktionalanalysis (Verlag Harri Deutsch)
• D. Werner: Funktionalanalysis (Springer Verlag)
• F. Hirzebruch, W. Scharlau: Einführung in die Funktionalanalysis (Vieweg)