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CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2019/2020

Advanced Analysis

The course is not on the list Without time-table
Code Completion Credits Range Language
A0B01PAN Z,ZK 6 2+2 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, 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:
http://math.feld.cvut.cz/veronika/vyuka/a0b01pan.htm
No time-table has been prepared for this course
The course is a part of the following study plans:
Data valid to 2019-09-22
For updated information see http://bilakniha.cvut.cz/en/predmet1830206.html