Functional Analysis 1
Code  Completion  Credits  Range  Language 

01FAN1  Z,ZK  4  2+2  Czech 
 Course guarantor:
 Pavel Šťovíček
 Lecturer:
 Pavel Šťovíček
 Tutor:
 Pavel Šťovíček
 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 HahnBanach 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, (McGrewHill, 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 timetable has been prepared for this course
 The course is a part of the following study plans: