Course detail

Functional Analysis II

FSI-SU2Acad. year: 2015/2016

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

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

doc. RNDr. Vítězslav Veselý, CSc.

Department

Institute of Mathematics (ÚM)

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.

Prerequisites

Differential and integral calculus. Basics in linear algebra, Fourier analysis and functional analysis.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

Course-unit credit will be awarded on the basis of individual in-depth study of a selected topic and by answering questions related to the theoretical background.

Course curriculum

Not applicable.

Work placements

Not applicable.

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.

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

A.E.Taylor: Úvod do funkcionální analýzy. Academia, Praha 1973. (CS)
A.N.Kolmogorov, S.V.Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975. (CS)
Ch.Heil: A Basis Theory Primer, expanded edition, Birkhäuser, New York, 2011. (EN)
L.Debnath, P.Mikusinski: Introduction to Hilbert spaces with Applications. 2-nd ed., Academic Press, London, 1999. (EN)

Recommended reading

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

Classification of course in study plans

  • Programme N3901-2 Master's

    branch M-MAI , 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Vítězslav Veselý, CSc.

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
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

doc. RNDr. Vítězslav Veselý, CSc.

Syllabus

Refreshing the knowledge acquired in the course Functional analysis I and practising the topics presented at the lectures using particular examples of commonly used functional spaces and operators.