The lecture deals with Banach and Hilbert spaces and the linear operators on these spaces. Typical examples are spaces of continuous or integrable functions, and linear operators on these spaces occur in the study of integral and differential equations. The development of functional analysis in the 20th century contributed significantly to the modern theory of differential equations. Today, functional analysis is a fundamental discipline of modern analysis and is widely used, for example, in the theory of partial differential equations, numerical mathematics, mathematical physics and many other areas of application.


Topics of the lecture:
basic properties and examples of metric spaces and Banach spaces, continuous linear operators on Banach spaces,  uniform boundedness principle, homomorphism theorem, Hilbert spaces, orthonormal bases, Sobolev spaces, dual spaces, Hahn-Banach theorem, weak convergence, Banach-Alaoglu theorem, reflexivity, compact linear operators

The contents of the basic lectures Analysis 1-3 and Linear Algebra 1+2 are assumed.

• D. Werner: Funktionalanalysis.

• H.W. Alt: Lineare Funktionalanalysis.
• H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations.
• J.B. Conway: A Course in Functional Analysis.
• M. Reed, B. Simon: Functional Analysis.
• W. Rudin: Functional Analysis.
• A.E. Taylor, D.C. Lay: Introduction to Functional Analysis.
• J. Wloka: Funktionalanalysis und Anwendungen.