Functional Analysis 2

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, Hilbert-Schmidt operators, spectral decomposition of bounded self-adjoint operators.

Requirements:

01FA1

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:

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, Hilbert-Schmidt operators.

6. Examples of the spectral decomposition of bounded self-adjoint 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, (McGrew-Hill, 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 time-table has been prepared for this course

The course is a part of the following study plans: