Functional Analysis II
FSI-SU2Acad. year: 2019/2020
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.
Learning outcomes of the course unit
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.
Differential and integral calculus. Basics in linear algebra, Fourier analysis and functional analysis.
Recommended optional programme components
Recommended or required reading
L.A.Ljusternik, V.J.Sobolev: Elementy funkcionalnovo analiza, (CS)
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. (CS)
J. Kačur: Vybrané kapitoly z matematickej fyziky I, skripta MFF UK, Bratislava 1984. (CS)
Ch.Heil: A Basis Theory Primer, expanded edition, Birkhäuser, New York, 2011. (EN)
A.E.Taylor: Úvod do funkcionální analýzy. Academia, Praha 1973. (CS)
A.W.Naylor, G.R.Sell: Teória lineárnych operátorov v technických a prírodných vedách, Alfa, Bratislava 1971 (CS)
A.N.Kolmogorov, S.V.Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975. (CS)
A.Ženíšek: Funkcionální analýza II, skripta FSI VUT, PC-DIR, Brno 1999. (CS)
L.Debnath, P.Mikusinski: Introduction to Hilbert spaces with Applications. 2-nd ed., Academic Press, London, 1999. (EN)
Planned learning activities and teaching methods
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.
Language of instruction
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.
Specification of controlled education, way of implementation and compensation for absences
Absence has to be made up by self-study and possibly via assigned homework.
Type of course unit
26 hours, optionally
Teacher / Lecturer
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 hours, compulsory
Teacher / Lecturer
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.