Functional Analysis 2
Code  Completion  Credits  Range  Language 

01FA2  Z,ZK  4  2+2  Czech 
 Course guarantor:
 Pavel Šťovíček
 Lecturer:
 Pavel Šťovíček
 Tutor:
 Pavel Šťovíček
 Supervisor:
 Department of Mathematics
 Synopsis:

The course aims to present selected fundamental results from functional analysis including basic theorems of the theory of Banach spaces, closed operators and their spectrum, HilbertSchmidt operators, spectral decomposition of bounded selfadjoint operators.
 Requirements:

01FA1
 Syllabus of lectures:

1. The Baire theorem, the BanachSteinhaus theorem (the principle of uniform boundedness), the open mapping theorem, the closed graph theorem.
2. Spectrum of closed operators in Banach spaces, the graph of an operator, analytic properties of a resolvent, the spectral radius.
3. Compact operators, the ArzelaAscoli theorem, HilbertSchmidt operators.
4. The Weyl criterion for normal operators, properties of spectra of bounded selfadjoint operators.
5, The spectral decomposition of bounded selfadjoint operators, functional calculus.
 Syllabus of tutorials:

1. Exercises devoted to basic properties of Hilbert spaces and to the orthogonal projection theorem.
2. The quotient of a Banach space by a closed subspace.
3. Properties of projection operators in Banach spaces and orthogonal projections in Hilbert spaces.
4. Examples of the application of the principle of uniform boundedness.
5. Exercises focused on integral operators, HilbertSchmidt operators.
6. Examples of the spectral decomposition of bounded selfadjoint operators.
 Study Objective:

Knowledge: Basics of the theory of Banach spaces, selected results about compact operators and the spectral analysis in Hilbert spaces.
Skills: Application of this knowledge in subsequent studies aimed at partial differential equations, integral equations, and problems of mathematical physics.
 Study materials:

Key references:
[1] J. Blank, P. Exner, M. Havlíček: Hilbert Space Operators in Quantum Physics, (American Institute of Physics, New York, 1994)
Recommended references:
[2] W. Rudin: Real and Complex Analysis, (McGrewHill, Inc., New York, 1974)
[3] A. N. Kolmogorov, S. V. Fomin: Elements of the Theory of Functions and Functional Analysis, (Dover Publications, 1999)
[4] 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: