Functional Analysis

Timetable

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.

Syllabus

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,  

 

examples.

 

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

 

theorem.

 

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

• B. Conway, Course in Functional Analysis, Springer-Verlag, New York 1997.
• Dunford, J.T. Schwartz, Linear Operators, Part I, Interscience Publishers, New York 1958.
• Goffman, G. Pedrick, First Course in Functional Analysis, Prentice-Hall Inc., Englewood Cliffs, New Jersey 1965.
• Rudin, Real and Complex Analysis, McGraw-Hill Inc., New York 1987.
• Rudin, Functional Analysis, McGraw-Hill Inc., New York 1991.