General information
| Course type | AMUPIE |
| Module title | Banach Spaces and Algebras |
| Language | English |
| Module lecturer | prof. UAM dr hab. Krzysztof Piszczek |
| Lecturer's email | kpk@amu.edu.pl |
| Lecturer position | Associate Professor |
| Faculty | Faculty of Mathematics and Computer Science |
| Semester | 2025/2026 (winter) |
| Duration | 60 |
| ECTS | 6 |
| USOS code | 06-DBSAUM0-E |
Timetable
2h lecture followed by 2h exercise classes - early hours preferably.
Module aim (aims)
Theory of Banach Spaces and Algebras is the language of modern analysis. It shows common roots of apparently quite different problems of classical analysis and it allows to solve many of them quite easily. It is necessary for proper understanding of the modern theory of differential equations, measure theory, harmonic analysis, and complex analysis among others. The aim of the course is to learn basic notions and tools like Banach and Hilbert spaces, linear operators, Banach Algebras, multiplicative functionals and and classical principles of this theory.
Pre-requisites in terms of knowledge, skills and social competences (where relevant)
Knowledge from mathematical analysis, elements of metric topology, and measure theory is advised for better understanding of the course.
Syllabus
Week 1: Banach Spaces - basic definitions and examples.
Week 2: Classical sequence and functions spaces: Hölder and Minkowski Inequalities.
Week 3: Linear and bounded operators, the operator norm, examples of linear bounded operators.
Week 4: Baire Category Theorem, the Banach-Steinhaus theorem, du Bois-Reymond’s theorem on convergence of Fourier series.
Week 5: The open mapping theorem, the Banach inverse mapping theorem, the closed graph theorem..
Week 6: The Hahn-Banach theorem.
Week 7: Duality theory for Banach Spaces.
Week 8: Inner products, unitary spaces, Hilbert spaces, examples.
Week 9: The orthogonal projection theorem, properties of orthogonal projections, the Riesz representation theorem.
Week 10: Orthonormal sets, Bessel’s inequality, Parseval’s identity, orthonormal basis, the Riesz-Fischer theorem.
Week 11: Banach Algebras: definitions, examples, constructions, basic properties.
Week 12: The group of invertible elements.
Week 13: The spectrum and spectral radius.
Week 14: Gelfand-Mazur Theorem and Gelfand-Beurling Formula.
Week 15: Holomorphic Functional Calculus.
Reading list
- B. Conway, Course in Functional Analysis, Springer-Verlag, New York 1997.
- W. Rudin, Functional Analysis, McGraw-Hill Inc., New York 1991.
- F. F. Bonsall, J. Duncan, “Complete Normed Algebras”, Springer, 1973.
- R. V. Kadison, J. R. Ringrose, “Fundamentals of the Theory of Operator Algebras”, vol. 1, AMS, 1997.