# 5 semester

 1 Name of the discipline FUNCTIONAL ANALYSIS 2 Course of Study specialty 3, Mathematics (scientific and pedagogical activity) 3 Semester of training 5 4 Amount of credits 3 5 Name of the lecturer Mazel Maya Khaimovna 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: 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 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. Theme 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, Fubini’s theorem. Theme 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. Theme 4. Continuous, uniformly continuous and Lipschitz maps. 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, laboratory practice, SSI (students scientific investigation) 11 Language of instruction Russian 12 Conditions (requirements), current control – Tests; – colloquium 13 Attestation form credit