1.       

Name of the discipline  

FUNCTIONAL ANALYSIS

2.       

Study course,

specialty

3, Computer mathematics and systems analysis

3.       

Semester of study

6

4.       

Amount of credits                    

2

5.       

Lecturer’s S.N.P.

Mazel Maya Khaimovna

6.       

Aims of the study

mastering the language of modern mathematics by students, mastering general constructs and the ability to apply them in theoretical and applied problems.

As a result of the study, the student should be able to:

be able to:

– Identify the design of functional analysis in specific problems;

– set properties of mappings in function spaces;

apply the results of functional analysis to solve theoretical and applied problems

7.       

Prerequisites

Theory of Probability, Mathematical Statistics, Equations of Mathematical Physics, Optimization Methods, Extremal Problems and Calculus of Variations, Numerical Methods

8.       

Contents of the discipline

Theme 5. Hilbert spaces. Definition of a scalar product. The Cauchy-Bunyakovskii inequality. Hilbert spaces. The projection theorem. The Fourier series theorem.

Theme 6. Linear equations in Banach spaces. Reversible operators. Theorems on invertibility. Banach’s theorem on the inverse operator. The spectrum and the resolution of a linear bounded operator.

Theme 7. Conjugate spaces and adjoint operators. Linear bounded functionals. The Hahn-Banach Theorem. General form of a linear bounded functional in Hilbert and some other specific spaces. The conjugate space. The adjoint operator and its properties. Theorem on the conditions for the solvability of the linear equation

Theme 8. Equations with compact operators. The Fredholm alternative for equations with operators of finite rank. Compact operators. Compactness of Integral Operators in Specific Spaces. Criterion for the finite-dimensionality of a normed space. The Riesz-Schauder theory for equations with compact operators in Hilbert space. The Fredholm alternative for integral equations

9.       

Recommended literature

Main literature:

1. Antonevich AB, Radyno Ya.V. Functional analysis and integral equations. 2 nd ed., Revised. and additional. Minsk, Publishing house of the Belarusian State University, 2006.

2. Antonevich AB, Mazel M.Kh., Radyno Ya.V. Functional analysis and integral equations. Tutorial. Minsk, Publishing house of the Belarusian State University, 2011.

3. Kolmogorov AN, Fomin S.V. Elements of the theory of functions and functional analysis. M., Fizmatlit, 2004.

4. Lyusternik LA, Sobolev VI Short course of functional analysis. M., Higher School, 1982.

5. Trenogin V.A. Functional analysis. M., Fizmatlit, 2002.

Additional literature:

1. Berezanskii Yu.M., Us G.Yu., Sheftel Z.G. Functional analysis. Lecture course. Kiev, High school, 1990.

2. Kantorovich LV, Akilov GP Functional analysis. St. Petersburg, Nevsky Dialect, BHV-Petersburg, 2002.

3. Kirillov AA, Gvishiani AD Theorems and problems of functional analysis. M., Science, 1979.

4. Antonevich AB, Knyazev PN, Radyno Ya.V. Tasks and exercises on functional analysis. Minsk, Higher School, 1978.

10.   

Methods of teaching

Lectures, laboratory practice, SSI (students scientific investigation)

11.   

Language of teaching

Russian

12.   

Knowledge control and requirements

– test;

– colloquium

The assessment on the exam is set taking into account: 30% – work in laboratory and practical classes, 70% – oral examination.

13.   

Certification form

examination