##### B232 - Summer 23/24

 Credits 6 Semesters Summer Completion Assessment + Examination Language of teaching Czech Extent of teaching 2P+2S
Annotation
Subject serves as an introduction to measure and integration theory and functional analysis. The first part deals with Lebesgue integration theory. Next parts are devoted to
basic concepts of the theory of Banach and Hilbert spaces and their connection to harmonic analysis. Last part deals with spectral theory of operators and their application to matrix analysis.
Course outlines
1. Measurable space. Field of measurable sets, measures.
2. Abstract Lebesgue integral and expectation value of random variable
3. Lebesgue measure in R^n (construction using outer measure). Lebesgue integral
4. Convergence theorems.
5. Product measure. Fubini Theorem
6. Integration in R^n - substitution theorem.
7. Normed space. Completeness. Bounded operators.
8. Inner product space. Hilbert space. projection Theorem.
9. Space L^2(R) as a Hilbert space. Density of smooth functions with compact support. Fourier transform in L^2(R). Plancherel Theorem.
10. Spectra of operators in a Hilbert space. Basic classes of operators in a Hilbert space - positive, self-adjoint, unitary, projection.
11. Diagonalization of a normal operator and matrix.
12. Decompositions of matrices and operators - spectral, polar, SVD.
13. Functions of operators and matrices.
14. Spare lecture
Exercises outlines
1. Measurable space. Field of measurable sets, measures.
2. Abstract Lebesgue integral and expectation value of random variable
3. Lebesgue measure in R^n (construction using outer measure). Lebesgue integral
4. Convergence theorems.
5. Product measure. Fubini Theorem
6. Integration in R^n - substitution theorem.
7. Normed space. Completeness. Bounded operators.
8. Inner product space. Hilbert space. projection Theorem.
9. Space L^2(R) as a Hilbert space. Density of smooth functions with compact support. Fourier transform in L^2(R). Plancherel Theorem.
10. Spectra of operators in a Hilbert space. Basic classes of operators in a Hilbert space - positive, self-adjoint, unitary, projection.
11. Diagonalization of a normal operator and matrix.
12. Decompositions of matrices and operators - spectral, polar, SVD.
13. Functions of operators and matrices.
14. Spare tutorial
Literature
[1] Rudin, W.: Analýza v reálném a komplexním oboru, Academia, 1977
[2] Kreyszig, E.: Introductory functional analysis with applications, Wiley 1989
[3] Lukeš, L.: Jemný úvod do funkcionální analýzy, Karolinum, 2005
[4] Meyer, C.D.: Matrix analysis and applied linear algebra, SIAM 2001.