Advanced Analysis
Code  Completion  Credits  Range  Language 

B0B01PAN  Z,ZK  6  2P+2S  Czech 
 Lecturer:
 Tutor:
 Supervisor:
 Department of Mathematics
 Synopsis:

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.
 Requirements:
 Syllabus of lectures:

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, selfadjoint, 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
 Syllabus of tutorials:

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, selfadjoint, 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
 Study Objective:
 Study materials:

[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.
 Note:
 Further information:
 http://math.feld.cvut.cz/veronika/vyuka/a0b01pan.htm
 No timetable has been prepared for this course
 The course is a part of the following study plans: