Functional analysis 2

Relations:

In order to register for the course 01FAN2, the student must have successfully completed or received credit for and not exhausted all examination dates for the course 01FANA1. The course 01FAN2 can be graded only after the course 01FANA1 has been successfully completed.

In order to register for the course 01FAN3, the student must have successfully completed or received credit for and not exhausted all examination dates for the course 01FAN2.

Course guarantor:

Lecturer:

Tutor:

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, Hilbert-Schmidt operators, spectral decomposition

of bounded self-adjoint operators.

Requirements:

Syllabus of lectures:

1. The Baire theorem, the Banach-Steinhaus 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 Arzela-Ascoli theorem, Hilbert-Schmidt operators.

4. The Weyl criterion for normal operators, properties of spectra of bounded self-adjoint operators.

5, The spectral decomposition of bounded self-adjoint operators, functional calculus.

Syllabus of tutorials:

Study Objective:

Study materials:

Povinná literatura

1. J. Blank, P. Exner, M. Havlíček: Hilbert Space Operators in Quantum Physics, Springer, 2013.

2. B. Simon: Operator Theory: A Comprehensive Course in Analysis, Part 4, AMS, Rhode Island, 2015.

Doporučená literatura

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:

• Aplikovaná algebra a analýza (compulsory course in the program)
• Matematické inženýrství - Matematická fyzika (PS)
• Matematické inženýrství - Matematická informatika (elective course)
• Matematické inženýrství - Matematické modelování (PS)
• Mathematical Engineering - Mathematical Physics (PS)