Functionalanalysis im Wintersemester 2025/2026

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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, called functionals. 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.

The first half of the topics of lecture will be applied to fundamental problems in data science: representation of models (neural networks itself are functionals in the above sense), reproducing kernel Hilbert spaces, universal approximation theorems (neural networks are dense in space of continuous functions), functional optimization (extreme value theory in infinite-dimensional spaces).
The topics of the second half of the lecture are applied to exemplary problems  or in the theory of partial differential equations and Sobolev spaces provding for instace a suitable solution framework beyond classical solutions to important equations from physics.

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.

Betreuung

Lecturer: Prof. Dr. André Schlichting

Teaching assistant: Nicolai Gerber

Vorlesungen
Fr. 12:00 bis 14:00 wöchentlich 17.10.2025 bis 13.02.2026 Raum: N24 - H12

Übungen
Fr. 12:00 bis 14:00 wöchentlich 17.10.2025 bis 13.02.2026 Raum: N24 - H12