5 semester

1.       

Name of the discipline

FUNCTIONAL ANALYSIS

2.       

Course of Study

specialty

3, Mathematics and Information Technology (mathematical and software of mobile devices)

3.       

Semester of training

5

4.       

Amount of credits

4

5.       

Name of the lecturer

Chesalin Vladimir Ivanovich

6.       

Objectives of studying the discipline

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:

–                    Identify the design of functional analysis in specific problems;

–                    to 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. Lebesgue measure. Systems of sets. Rings of sets, semirings, algebras, sigma-rings and sigma algebras, Borel sets. General concept of measure. Sigma-additive measure. Continuation of the Lebesgue measure. An exterior measure, measurable sets, sets of measure zero, the main theorem of measure theory. Lebesgue measure on a line. Lebesgue-Stieltjes measure.

Topic 2. The Lebesgue integral. Measurable functions, simple functions, integral of a simple function, integral of a measurable function, simplest properties of the Lebesgue integral, passage to the limit under the Lebesgue integral sign. The product of measures, The Fubini theorem.

Topic 3. Metric spaces. Definition and basic examples of functional metric spaces. Topology of metric spaces. Complete metric spaces. Replenishment of metric spaces. Compact metric spaces.

Topic 4. Continuous, uniformly continuous and Lipschitz mappings. Definitions and properties. The continuation theorem. Application of the principle of contracting mappings to 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.   

Teaching Methods

Lectures, practical classes, training and research

11.   

Language of instruction

Russian

12.   

Conditions (requirements),

current control

– Test;

– Colloquium

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

13.   

Attestation form

Test, exam