1.       

Name of the discipline  

FUNCTIONAL ANALYSIS

2.       

Study course,

specialty

3, Computer mathematics and systems analysis

3.       

Semester of study

5

4.       

Amount of credits                    

3

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

Topic 1. Metric spaces. Metric spaces. Topolgy, generated by the metric. Basic examples of functional metric spaces. Complete spaces. The completion theorem.

Theme 2. Continuous, uniformly continuous and Lipschitz maps. Theorems on continuation. The principle of contracting mappings and its application to integral equations.

Theme 3. Measure and Lebesgue integral. Systems of subsets: rings, algebras, sigma algebras. General concept of measure. Sigma-additive measures Continuation of the Lebesgue measure. The main theorem. The Lebesgue measure and the Lebesgue-Stieltjes measure on the line. Measurable functions, simple functions. The integral of a simple function. General definition of the Lebesgue integral. Theorems on the passage to the limit under the integral sign. Inequalities of Jung, Hölder, Minkowski.

Topic 4. Normalized spaces. Vector, normalized, Banach spaces. Series in Banach spaces. Linear operators. The norm of a bounded operator. The space of linear bounded operators. The Banach-Steinhaus theorem

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

Tests

colloquium

13.   

Certification form

credit