Functional Analysis II (FSI-SU2)

Academic year 2020/2021
Supervisor: prof. Aleksandre Lomtatidze, DrSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
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.
Learning outcomes and competences:
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.
Prerequisites:
Differential and integral calculus. Basics in linear algebra, Fourier analysis and functional analysis.
Course contents:
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.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focussed on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
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.
Controlled participation in lessons:
Absence has to be made up by self-study and possibly via assigned homework.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture 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 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.
Literature - fundamental:
1. 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.
2. Ch.Heil: A Basis Theory Primer, expanded edition, Birkhäuser, New York, 2011.
3. A.E.Taylor: Úvod do funkcionální analýzy. Academia, Praha 1973.
4. A.N.Kolmogorov, S.V.Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975.
5. L.Debnath, P.Mikusinski: Introduction to Hilbert spaces with Applications. 2-nd ed., Academic Press, London, 1999.
Literature - recommended:
1. L.A.Ljusternik, V.J.Sobolev: Elementy funkcionalnovo analiza,
2. J. Kačur: Vybrané kapitoly z matematickej fyziky I, skripta MFF UK, Bratislava 1984.
4. A.W.Naylor, G.R.Sell: Teória lineárnych operátorov v technických a prírodných vedách, Alfa, Bratislava 1971
5. A.Ženíšek: Funkcionální analýza II, skripta FSI VUT, PC-DIR, Brno 1999.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
M2A-P full-time study M-MAI Mathematical Engineering -- Cr,Ex 3 Compulsory 2 1 W