General information

Course type AMUPIE
Module title Functional Analysis
Language English
Module lecturer prof. UAM Krzysztof Piszczek
Lecturer's email
Lecturer position Associate Professor
Faculty Faculty of Mathematics and Computer Science
Semester 2023/2024 (winter)
Duration 60
USOS code 06-DANFUM0-E


1,5h lecture followed by 1,5h exercise classes - preferably on Monday.

Module aim (aims)

Functional analysis is the language of the 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 the modern theory of differential equations, measure theory, harmonic analysis, and complex analysis. The aim of the course is to learn basic notions and tools like Banach and Hilbert spaces, linear operators, and basic theorems like the Banach-Steinhaus theorem, the open mapping theorem, the closed graph theorem, and the Hahn-Banach theorem.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

One has to know a course of mathematical analysis, elements of metric topology, and measure theory.


Week 1: Norms, normed spaces, Banach spaces. Examples of Banach spaces (spaces of continuous functions,  the space of continuously differentiable functions, the space of bounded sequences, and the space of zero-convergent sequences).


Week 2: Spaces of integrable functions, Hölder’s and Minkowski’s inequalities, L_p norms, completeness.


Week 3: Series in normed spaces, separable Banach spaces, examples.


Week 4: Linear and bounded operators, the operator norm, examples of linear bounded operators.


Week 5: The space of linear bounded operators between normed spaces, finite dimensional normed spaces.


Week 6: Baire’s theorem, the Banach-Steinhaus theorem, du Bois-Reymond’s theorem on convergence of Fourier series.


Week 7: The open mapping theorem, the Banach inverse mapping theorem, the closed graph theorem.


Week 8: The Hahn-Banach theorem.


Week 9: Consequences of the Hahn-Banach theorem, the dual space to a Banach space, examples.


Week 10: Inner products, the norm induced by an inner product,  unitary spaces, Hilbert spaces,  




Week 11: The orthogonal projection theorem, properties of orthogonal projections, the Riesz


representation theorem.


Week 12: Orthonormal sets, Bessel’s inequality, Parseval’s identity, orthonormal basis, the Riesz-Fischer




Week 13: Compactness in normed spaces, the Hausdorff theorem, the Arzelà-Ascoli theorem.


Week 14: The Riesz lemma, compactness of the unit ball in a normed space, definition and basic


properties of compact operators.          


Week 15: Examples of compact operators. Integral operators.     

Reading list