Subject description - A0B01PAN

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A0B01PAN Advanced Analysis Extent of teaching:2+2
Guarantors:Hamhalter J. Roles:V Language of
teaching:
CS
Teachers:Hamhalter J., Sobotíková V. Completion:Z,ZK
Responsible Department:13101 Credits:6 Semester:L

Anotation:

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 outline:

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.

Requirements:

Předmět je zakončen standardně zápočtem a zkouškou. Podmínkou pro získání zápočtu je aktivní účast na výuce. Hodnocení předmětu bude záviset na zkoušce samotné. Zkouška je ústní a je při ní zkoušena probraná látka. Další informace viz http://math.feld.cvut.cz/veronika/vyuka/a0b01pan.htm

Webpage:

http://math.feld.cvut.cz/veronika/vyuka/a0b01pan.htm

Subject is included into these academic programs:

Program Branch Role Recommended semester
BPOI1 Computer Systems V
BPOI_BO Common courses V
BPOI3 Software Systems V
BPOI2 Computer and Information Science V
BPKYR1 Robotics V
BPKYR_BO Common courses V
BPKYR3 Systems and Control V
BPKYR2 Sensors and Instrumentation V
BPKME1 Communication Technology V
BPKME5 Komunikace a elektronika V
BPKME_BO Common courses V
BPKME4 Network and Information Technology V
BPKME3 Applied Electronics V
BPKME2 Multimedia Technology V
BPEEM1 Applied Electrical Engineering V
BPEEM_BO Common courses V
BPEEM2 Electrical Engineering and Management V
BMI(ECTS) Manager Informatics V
BWM(ECTS) Web and Multimedia V
BIS(ECTS) Intelligent Systems V
BSI(ECTS) Software Engineering V


Page updated 27.5.2019 07:53:08, semester: Z,L/2020-1, L/2019-20, Z,L/2018-9, Z/2019-20, Send comments about the content to the Administrators of the Academic Programs Proposal and Realization: I. Halaška (K336), J. Novák (K336)