Functional analysis is the study of infinite-dimensional topological vector spaces (e.g., normed vector spaces) and the continuous linear maps between them. The subject involves a rich mixture of topology, measure, algebra, and combinatorics and plays an important role in many areas of pure and applied mathematics, from differential equations and probability to (noncommutative) geometry, ergodic theory, and topology. Topics to be covered include Banach spaces, the Hahn-Banach theorem, the closed graph and open mapping theorems, the uniform boundedness principle, duality, operators on Hilbert space, the spectral theory of compact operators, and the Fredholm index. The course forms the second part of the Vertiefungsmodul "Funktionalanalysis" and can be taken as preparation for a Bachelor-Arbeit in the area of functional analysis. A second course in functional analysis treating more specialized topics will be offered in the winter 2022 semester.

Resources:

• D. Werner, Funktionalanalysis
• G. K. Petersen, Analysis Now
• J. B. Conway, A Course in Functional Analysis
• W. Rudin, Functional Analysis
• F. Hirzebruch und W. Scharlau, Einführung in die Funktionalanalysis
• G. B. Folland, Real Analysis
• G. J. Murphy, C*-Algebras and Operator Theory

More information will be posted in Learnweb.