Course detail
Mathematics - Selected Topics I
FSI-T1K Acad. year: 2023/2024 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.
Supervisor
Department
Learning outcomes of the course unit
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
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 oral part.
Language of instruction
Czech
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.
Specification of controlled education, way of implementation and compensation for absences
Missed lessons can be compensated for via a written test.
The study programmes with the given course
Programme B-FIN-P: Physical Engineering and Nanotechnology, Bachelor's
branch ---: no specialisation, 3 credits, compulsory
Type of course unit
Lecture
26 hours, optionally
Teacher / Lecturer
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
Teacher / Lecturer
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