Functional analysis in winter 2023/2024

Please register in Moodle for the course. All up to date information will be provided there.

The foundational idea of functional analysis is to interpret sequences or functions as points in a suitable vector space and to consider problems in Analysis by studying mappings on such spaces. Nontrivial statements can be derived by endowing these spaces with a norm and by studying analytical properties such as continuity of these mappings. 

The lecture covers the fundamental notions of Banach spaces, Hilbert spaces and linear operators. Central topics within this theory are the Hahn-Banach and the Lax-Milgram theorem, the open mapping theorem and the closed graph theorem.

Moreover, we will study properties of compact operators, reflexive spaces and discuss Fredholm theory. Then, the spectral theorem will be presented.

All topics of the lecture will be applied to exemplary problems, for instance in data science or in the theory of partial differential equations and Sobolev spaces.

This module counts as an elective in pure math for both bachelor and master programs in Maths, Mathematics and Management as well as Mathematical Biometry.

Moreover, the first half of this lecture is mandatory for students in Mathematical Data Science.

• H. W. Alt, Lineare Funktionalanalysis, Springer 2012
• D. Werner: Funktionalanalysis, Springer 2011
• P. D. Lax: Functional analysis, Wiley 2002
• H. Brézis: Functional analysis, Sobolev spaces and partial differential equations, Springer 2011
• J. B. Conway: A Course in Functional Analysis, Springer 2010

Diese und weitere Bücher findet man auch im Semesterapparat von Prof. Dr. Dall'Acqua.