Mathematics - Selected Topics I (FSI-T1K)

Academic year 2023/2024
Supervisor: prof. RNDr. Miloslav Druckmüller, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:

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.
Learning outcomes and competences:

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.
Prerequisites:
Real and complex analysis
Course contents:
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.
Teaching methods and criteria:
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 oral part.
Controlled participation in lessons:
Missed lessons can be compensated for via a written test.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture

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

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
Literature - fundamental:
1. Kolmogorov,A.N.,Fomin,S.V.: Základy teorie funkcí a funkcionální analýzy, SNTL Praha 1975
2. Lang, S. Real and Functional Analysis. Third Edition, Springer-Verlag 1993
Literature - recommended:
1. Kolmogorov,A.N.,Fomin,S.V.: Základy teorie funkcí a funkcionální analýzy, SNTL Praha 1975
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B-FIN-P full-time study --- no specialisation -- Cr,Ex 3 Compulsory 1 2 S