Course detail

Functional Analysis II

FSI-SU2-AAcad. year: 2024/2025

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

Mode of study

Not applicable.

Guarantor

prof. Aleksandre Lomtatidze, DrSc.

Department

Institute of Mathematics (ÚM)

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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

A.E.Taylor: Úvod do funkcionální analýzy. Academia, Praha 1973.
A.N.Kolmogorov, S.V.Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975.
Ch.Heil: A Basis Theory Primer, expanded edition, Birkhäuser, New York, 2011.
L.Debnath, P.Mikusinski: Introduction to Hilbert spaces with Applications. 2-nd ed., Academic Press, London, 1999.
V. Veselý a P. Rajmic. Funkcionálnı́ analýza s aplikacemi ve zpracovánı́ signálů. Odborná učebnice. Vysoké učenı́ technické v Brně, Brno (CZ), 2015. ISBN 978-80-214-5186-5.

Recommended reading

A.W.Naylor, G.R.Sell: Teória lineárnych operátorov v technických a prírodných vedách, Alfa, Bratislava 1971
A.Ženíšek: Funkcionální analýza II, skripta FSI VUT, PC-DIR, Brno 1999.
J. Kačur: Vybrané kapitoly z matematickej fyziky I, skripta MFF UK, Bratislava 1984.
L.A.Ljusternik, V.J.Sobolev: Elementy funkcionalnovo analiza,

Classification of course in study plans

  • Programme N-AIM-A Master's 2 year of study, winter semester, compulsory
  • Programme N-MAI-A Master's 1 year of study, winter semester, compulsory

  • Programme C-AKR-P Lifelong learning

    specialization CZS , 1 year of study, winter semester, elective

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. Aleksandre Lomtatidze, DrSc.

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 hod., compulsory

Teacher / Lecturer

prof. Aleksandre Lomtatidze, DrSc.

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.