Course detail

Mathematics - Selected Topics I

FSI-T1KAcad. year: 2024/2025

The course includes selected topics of functional analysis which are necessary for application in physics. It focuses on functional spaces, orthogonal systems and orthogonal transformations.

Language of instruction

Czech

Number of ECTS credits

3

Mode of study

Not applicable.

Guarantor

prof. RNDr. Miloslav Druckmüller, CSc.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Real and complex analysis

Rules for evaluation and completion of the course

Course-unit credit - based on a written test
Exam has a written and oral part.


Missed lessons can be compensated for via a written test.

Aims

The aim of the course is to extend students´ knowledge in algebra and analysis acquired in the basic mathematical course by the topics necessary for study of physical engineering.


Basic knowledge of functional analysis, metric, vector and unitary spaces, Hilbert space, orthogonal systems of functions, orthogonal transforms, Fourier transform and spectral analysis, application of the mentioned subjects in physics.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Kolmogorov,A.N.,Fomin,S.V.: Základy teorie funkcí a funkcionální analýzy, SNTL Praha 1975 (CS)
Lang, S. Real and Functional Analysis. Third Edition, Springer-Verlag 1993 (EN)

Recommended reading

Kolmogorov,A.N.,Fomin,S.V.: Základy teorie funkcí a funkcionální analýzy, SNTL Praha 1975

Classification of course in study plans

  • Programme B-FIN-P Bachelor's 2 year of study, summer semester, compulsory

  • Programme C-AKR-P Lifelong learning

    specialization CLS , 1 year of study, summer semester, elective

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

Syllabus

1. Relations, equivalence, factor set, group, isomorphism
2. Metric space, complete metric space
3. Contraction, fix-point Banach's theorem and its applications
4. Vector space, base, dimension, isomorphism
5. Automorphism of vector spaces, eigenvectors and eigenvalues
6. Normed space, Unitary space orthogonal a orthonormal bases
7. Orthogonal a orthonormal bases, isomorphism
8. Hilbert space, isomorphism, L2 and l2 spaces
8. Orthogonal bases, Fourier series
10. Complex Fourier series, discrete Fourier transform
11. Usage of Fourier transform, convolution theorem
12. L2 space for functions of more variable
13. Operators and functionals in Hilbert space

Exercise

13 hod., compulsory

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

Syllabus

1. Relations, equivalence, factor set, group, isomorphism
2. Metric space, complete metric space
3. Contraction, fix-point Banach's theorem and its applications
4. Vector space, base, dimension, isomorphism
5. Automorphism of vector spaces, eigenvectors and eigenvalues
6. Normed space, Unitary space orthogonal a orthonormal bases
7. Orthogonal a orthonormal bases, isomorphism
8. Hilbert space, isomorphism, L2 and l2 spaces
8. Orthogonal bases, Fourier series
10. Complex Fourier series, discrete Fourier transform
11. Usage of Fourier transform, convolution theorem
12. L2 space for functions of more variable
13. Operators and functionals in Hilbert space