Course detail
Mathematics - Selected Topics I
FSI-T1K Acad. year: 2024/2025 Summer semester
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
Supervisor
Department
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.
The study programmes with the given course
Programme B-FIN-P: Physical Engineering and Nanotechnology, Bachelor's, compulsory
Programme C-AKR-P: , Lifelong learning
specialization CLS: , elective
Type of course unit
Lecture
26 hours, optionally
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 hours, compulsory
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