Course detail
Functional Analysis II
FSI-SU2-A Acad. year: 2024/2025 Winter semester
Review of topics presented in the course Functional Analysis I.
Theory of bounded linear operators. Compact sets and operators.
Inverse and pseudoinverse of bounded linear operators.
Bases primer: orthonormal bases, Riesz bases and frames.
Spectral theory of self-adjoint compact operators.
Language of instruction
English
Number of ECTS credits
4
Supervisor
Department
Entry knowledge
Differential and integral calculus. Basics in linear algebra, Fourier analysis and functional analysis.
Rules for evaluation and completion of the course
Course-unit credit will be awarded on the basis of student's activity in tutorials focussed on solving tasks/problems announced by the teacher, and/or alternatively due to an idividual in-depth elaboration of selected topic(s).
The attendance in tutorials is compulsory. Examinations at a regular date are written or oral, the examinations at a resit or alternative date oral only. Examinations assess student's knowledge of the theoretical background an his/her ability to apply acquired skills independently and creatively.
Absence has to be made up by self-study and possibly via assigned homework.
Aims
The aim of the course is to make students familiar with main results of linear functional analysis and their application to solution of problems of mathematical modelling.
Knowledge of basic topics of functional analysis, of the theory of function spaces and linear operators. Problem solving skill mainly in Hilbert spaces, solution by means of abstract Fourier series and Fourier transform.
The study programmes with the given course
Programme N-AIM-A: Applied and Interdisciplinary Mathematics, Master's, compulsory
Programme N-MAI-A: Mathematical Engineering, Master's, compulsory
Type of course unit
Lecture
26 hours, optionally
Syllabus
1. Review: topological, metric, normed linear and inner-product spaces, revision, direct product and factorspace
2. Review: dual spaces, continuous linear functionals, Hahn-Banach theorem, weak convergence
3. Review: Fourier series, Fourier transform and convolution
4. Bounded linear operators and associated main results
5. Adjoint and self-adjoint operatots incl. othogonal projection
6. Riesz Representation Theorem and Banach-Steinhaus Theorem
7. Unitary operators, compact sets and compact operators
8. Inverse of bounded linear operators in Banach and Hilbert spaces
9. Pseudoinverse of bounded linear operators in Hilbert spaces
10. Bases primer: orthonormal bases, Riesz bases and frames
11. Spectral theory of self-adjoint compact operators, Hilbert-Schmidt Theorem
12. Examples and applications primarily related to the field of Fourier analysis and signal processing
13. Reserve
Exercise
13 hours, compulsory
Syllabus
Refreshing the knowledge acquired in the course Functional analysis I and practising the topics presented at the lectures by individually solving selected examples and/or problems.