Functional Analysis 1

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

Basic notions and results are addressed concerning successively topological spaces, metric spaces, topological vector spaces, normed and Banach spaces, Hilbert spaces.

Requirements:

The complete introductory course in mathematical analysis and linear algebra on level A or B given at the Faculty of Nuclear Sciences and Physical Engineering

Syllabus of lectures:

1. Topological spaces

2. Metric spaces, compactness criteria, completion of a metric space

3. Topological vector spaces

4. Minkowski functional, the Hahn-Banach theorem

6. Metric vector spaces, Fréchet spaces

6. Normed vector spaces, bounded linear mappings, the operator norm

7. Banach spaces, extension of a bounded operator

8. Banach spaces of integrable functions

9. Hilbert spaces, orthogonal projection, orthogonal basis

10. The Riesz representation theorem, adjoint operator

Syllabus of tutorials:

Exercise is closely linked to the lecture, which is illustrated by appropriate examples. Accent is placed on the correctness of the calculation.

1. Basics of topology, repetition.

2. Basics of metric spaces and of banach spaces.

3. Banach spaces and linear bounded mappings

4. Resolvent formula, Fourier transform

5. Scalar product, isomorphism of Hilbert spaces orthogonality

7. Norms, continuity, linear extension, projectors, types of convergence

8. Spectral properties of normal and compact operators, ideals of compact operators

Study Objective:

Knowledge: basics of Banach and Hilbert spaces and linear operators in these spaces, and as a background sufficiently profound knowledge of topological and metric spaces

Skills: applications of the apparatus of Banach and Hilbert spaces

Study materials:

Key references:

[1] J.Blank.P,Exner,M.Havlíček: Hilbert Space Operators in Quantum Physics, Springer,2008.

Recommended references:

[2] M. Reed, B. Simon : Methods of Modern Mathematical Physics I.. ACADEMIC PRESS, N.Z. 1972

[3] W. Rudin: Real and Complex Analysis, (McGrew-Hill, Inc., New York, 1974)

[4] A. N. Kolmogorov, S. V. Fomin: Elements of the Theory of Functions and Functional Analysis, (Dover Publications, 1999)

[5] A. E. Taylor: Introduction to Functional Analysis, (John Wiley and Sons, Inc., New York, 1976)

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• BS Matematické inženýrství - Matematické modelování (compulsory course of the specialization, elective course)
• BS Matematické inženýrství - Matematická fyzika (compulsory course of the specialization, elective course)
• BS Matematické inženýrství - Aplikované matematicko-stochastické metody (elective course)
• BS Informatická fyzika (elective course)
• BS Aplikace softwarového inženýrství (elective course)
• BS Aplikovaná informatika (elective course)
• BS jaderné inženýrství B (elective course)
• BS Jaderné inženýrství C (elective course)
• BS Dozimetrie a aplikace ionizujícího záření (elective course)
• BS Experimentální jaderná a částicová fyzika (elective course)
• BS Inženýrství pevných látek (elective course)
• BS Diagnostika materiálů (elective course)
• BS Fyzika a technika termojaderné fúze (elective course)
• BS Fyzikální elektronika (elective course)
• BS Jaderná chemie (elective course)