# 5 semester

 1 Name of the discipline FUNCTIONAL ANALYSIS 2 Course of Study specialty 3, Mechanics and mathematical modeling 3 Semester of training 5 4 Amount of credits 3 5 Name of the lecturer Chesalin Vladimir Ivanovich 6 Objectives of studying the discipline Mastering students with the language of modern mathematics, mastering general constructs and their ability to apply them in theoretical and applied problems. Familiarization of students with the basic principles of functional analysis and examples of their applications. Further development of students’ skills in abstract mathematical thinking and the ability to apply it in specific tasks, increase their mathematical culture. As a result of the study, the student should 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 Algebra and number theory, Discrete Math, Analytic geometry, Mathematical analysis, Differential equations, Theory of functions of a complex variable 8 Contents of the discipline Section 1. METRIC SPACES. Topic 1.1. Metric spaces. The topology generated by the metric. Basic examples of functional metric spaces. Complete spaces. The completion theorem. Section 2. CONTINUOUS, UNIFORM CONTINUOUS AND LIPSZITS DISPLAYS. Topic 2.1. Compressive mapping. Theorems on continuation. The principle of contracting mappings and its application to integral equations. Section 3. MEASURE AND LEBEGE INTEGRAL. Topic 3.1. General concept of measure. The problem of completion of the space of continuous functions with integral metric. Systems of subsets: rings, algebras, -algebras. Topic 3.2. -additive measures. -additivity of length. 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. Topic 3.3. Theorems on the passage to the limit under the integral sign. Inequalities of Jung, Hölder, Minkowski. Spaces , their completeness. Application of the principle of contracting mappings to integral equations in . Section 4. NORMED SPACES. Topic 4.1. Vector, normed, Banach spaces. Series in Banach spaces. Linear operators. The norm of a bounded operator. The space of linear bounded operators. The Banach-Steinhaus theorem. Section 5. HILBERT SPACES. Topic 5.1. Hilbert spaces. Definition of a scalar product. The Cauchy-Bunyakovskii inequality. The projection theorem. The Fourier series theorem. Section 6. LINEAR EQUATIONS IN BANACH SPACES. Topic 6.1. Invertible operators. Theorems on invertibility. Topic 6.2. Spectrum and resolution of a linear bounded operator. The Banach theorem on the inverse operator. Section 7. CONJUGATED SPACES AND CONJUGATED OPERATORS. Topic 7.1. Linear bounded functionals. The Hahn-Banach Theorem. General form of a linear bounded functional in Hilbert and some other specific spaces. Topic 7.2. Conjugate spaces and operators. The theorem on the conditions for the solvability of a linear equation. Section 8. EQUATIONS WITH COMPACT OPERATORS. Theme 8.1. Compact operators. The Fredholm alternative for equations with operators of finite rank. Compactness of Integral Operators in Specific Spaces. Criterion for the finite-dimensionality of a normed space. Topic 8.2. The Riesz-Schauder theory for equations with compact operators in Hilbert space. The Fredholm alternative for integral equations in spaces  and . 9 Recommended literature Main literature: 1. Antonevich, AB Functional Analysis and Integral Equations: A Textbook / AB Antonevich, Ya. V. Radyno. 2 nd ed., Revised. and additional. – Minsk: BSU, 2006. – 430 p. 2. Antonevich, AB Functional Analysis and Integral Equations: Proc. allowance / AB Antonevich, M.H. Mazel Ya. V. Radyno. – Minsk: BSU, 2011. – 319 with. 3. Kolmogorov, AN Elements of the theory of functions and functional analysis / AN Kolmogorov, SV Fomin. 7 th ed. – M.: Fizmatlit, 2004. – 572 p. 4. Lyusternik, LA A short course of functional analysis / LA Lyusternik, VI Sobolev. 2 nd ed., Sr. – M.: Fizmatlit, 2009. – 272 p. 5. Trenogin, VA A. Functional analysis / VA Trenogin. 3rd ed., Rev. – M.: Fizmatlit, 2002. – 488 p. Additional literature: 6. Berezanskii, Yu. M. Functional Analysis. Course of lectures: Textbook. allowance / Yu. M. Berezanskii, GF Us, ZG Sheftel. – К.: Выща школа, 1990. – 600 с. 7. Vorovich, II, Functional Analysis and Its Applications in Continuum Mechanics: Proc. allowance / II Vorovich, L. P. Lebedev. 3rd ed., Rev. – M.: The University Book, 2000. – 488 p. 8. Kirillov, AA Theorems and problems of functional analysis / AA Kirillov, AD Gvishiani. – M.: Science, 1979. – 488 p. 9. Antonevich, AB Problems and exercises in functional analysis: Textbook. allowance for math. specialist. universities / AB Antonevich, PN Knyazev, YV Radyno; Ed. S. G. Kreina – Minsk: Vysh. school, 1978. – 208 p. 10 Teaching Methods Interactive teaching methods (working in small groups (team), problem-based learning) is organized taking into account the inclusion in the learning process of all students of the group. Joint activity means that each student makes his own individual contribution, in the course of the work there is an exchange of knowledge, ideas, methods of activity. Organized individual, steam and group work. Interactive methods are based on the principles of interaction, activity of trainees, reliance on group experience, mandatory feedback 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 Exam