Course detail

Mathematics - Selected Topics I

FSI-T1KAcad. year: 2022/2023

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)

Learning outcomes of the course unit

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

Prerequisites

Real and complex 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 - based on a written test
Exam has a written and an oral part.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

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

Specification of controlled education, way of implementation and compensation for absences

Missed lessons can be compensated for via a written test.

Recommended optional programme components

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

Elearning

eLearning: currently opened course

Classification of course in study plans

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

  • Programme LLE Lifelong learning

    branch CZV , 1 year of study, summer semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

Syllabus

1. Introduction
2. Metric space
3. Contraction, fix point Banach's theorem
4. Vector space, base, dimension, Vector spaces of functions
5. Unitary space orthogonal a orthonormal spaces
6. Hilbert space, L2 and l2 space
7. Orthogonal bases, Fourier series
8. Orthogonal transforms, Fourier transform
9. Usage of Fourier transform, convolution theorem
10.2D Fourier transform
11.Filtration in space and frequency domain, applications in physics
12. Operators and functionals
13. Variation methods

Exercise

13 hod., compulsory

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

Syllabus

1. Introduction
2. Metric space
3. Fix point Banach's theorem applications
4. Vector space, base, dimension, Vector spaces of functions
5. Unitary space orthogonal a orthonormal spaces
6. Hilbert space, L2 and l2 space
7. Orthogonal bases, Fourier series
8. Orthogonal transforms, Fourier transform
9. Usage of Fourier transform, convolution theorem
10. 2D Fourier transform
11. Filtration in space and frequency domain, applications in physics
12. Operators and functionals
13. Variation methods

Elearning

eLearning: currently opened course