Functional analysis 2

Lecturer:

Pavel Šťovíček (gar.)

Tutor:

Pavel Šťovíček (gar.)

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, Hilbert-Schmidt operators, spectral decomposition of bounded self-adjoint operators and basics of the theory of unbounded self-adjoint operators.

Requirements:

Basic courses of Calculus and Linear Algebra, and Functional Analysis 2 (in the extent of the courses 01MA1, 01MAA2-4, 01LA1, 01LAA2, 01FA1 held at the FNSPE CTU in Prague).

Syllabus of lectures:

1. The Baire theorem, the Banach-Steinhaus theorem (the principle of uniform boundedness), the open mapping theorem, the closed graph theorem, the Hahn-Banach 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 (a summary), the Arzela-Ascoli theorem, Hilbert-Schmidt operators. 4. The Weyl criterion, properties of spectra of bounded self-adjoint operators and the spectral decomposition, functional calculus. 5. Adjoint operators to unbounded operators in Hilbert spaces, basics of the theory of self-adjoint extensions of symmetric operators.

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 of basics of the theory of Banach spaces, selected results about compact operators and the spectral analysis in Hilbert spaces. Skills to apply this knowledge during subsequent studies aimed on partial differential equations and integral equations, and in solving 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:

Time-table for winter semester 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans: