Course detail

Functional Analysis II

FSI-SU2Acad. year: 2011/2012

Revision of topics presented in the course Functional Analysis I. Functionals, dual spaces. Theory of linear operators. Compact sets and operators. Inversion of bounded linear operators. Spectral theory of compact operators.

Language of instruction

Czech

Number of ECTS credits

4

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 of functional analysis, differential equations.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

Assesment methods and criteria linked to learning outcomes

Course-unit credit is awarded on condition of having attended the seminars actively. Oral examination tests the knowledge of definitions and includes questions regarding the theory.

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)
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. Topological, metric, normed linear and inner-product spaces, revision.
2. Projection, factorspace.
3. Orthogonal basis, abstract Fourier series.
4. Continuous linear functionals, Hahn-Banach theorem.
5. Weak convergence.
6. Linear operators, Banach-Steinhaus theorem.
7. Adjoint operator.
8. Compact sets and compact operators.
9. Inversion of linear operators in Banach and Hilbert spaces.
10. Spectral theory of linear compact operators.
11. Hilbert-Schmidt theorem.
12. Examples and applications.
13. Reserve.

Exercise

13 hod., compulsory

Teacher / Lecturer

doc. RNDr. Vítězslav Veselý, CSc.

Syllabus

Revision of the knowledge acquired in the course Functional analysis I and practising the topics presented at the lectures using particular examples of spaces of sequences and spaces of integrable function.