Advanced Analysis

Course guarantor:

Jan Hamhalter

Lecturer:

Jan Hamhalter, Veronika Sobotíková

Tutor:

Jan Hamhalter, Veronika Sobotíková

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, 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

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, 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

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:

https://moodle.fel.cvut.cz/courses/B0B01PAN

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
Tue
Wed
Thu
Fri	roomT2:C3-51 Hamhalter J. Sobotíková V. 11:45–13:30 (lecture parallel1) Dejvice roomT2:C3-51 Hamhalter J. Sobotíková V. 13:30–15:15 (lecture parallel1 parallel nr.101) Dejvice roomT2:C2-83 Hamhalter J. Sobotíková V. 11:45–13:30 (lecture parallel1) Dejvice roomT2:C2-83 Hamhalter J. Sobotíková V. 13:30–15:15 (lecture parallel1 parallel nr.101) Dejvice

The course is a part of the following study plans:

• Cybernetics and Robotics 2016 (compulsory elective course)