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Title |
FUNCTIONAL ANALYSIS |
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Course |
2, Mathematics (research and development) |
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3 |
Semester of training |
4 |
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4 |
Credits |
3 |
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5 |
Lecturer |
Yablonskaya Anna Gennadevna |
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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 studying the academic discipline, 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; |
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Requirements |
Algebra and number theory, Discrete Math, Analytic geometry, Mathematical analysis, Differential equations, Theory of functions of a complex variable |
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8 |
Contents of the discipline |
Theme 1 Metric spaces. Metric spaces. The topology generated by the metric. Basic examples of functional metric spaces. Complete spaces. The completion theorem. Theme 2 Continuous, uniformly continuous and Lipschitz mappings. Theorems on continuation. The principle of contracting mappings and its application to integral equations. Topic 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. Theme 4 Normalized spaces and linear operators. 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 |
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9 |
References |
Main literature: 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. Antonevich AB, Mazel M.Kh., Radyno Ya.V. Functional analysis and integral equations. Tutorial. Minsk, Publishing house of the Belarusian State University, 2011. Kolmogorov AN, Fomin S.V. Elements of the theory of functions and functional analysis. M., Fizmatlit, 2004. Lyusternik LA, Sobolev VI Short course of functional analysis. M., Higher School, 1982. Trenogin V.A. Functional analysis. M., Fizmatlit, 2002. Additional literature: Berezanskiy Yu.M., Us G.Yu., Sheftel Z.G. Functional analysis. Lecture course. Kiev, High school, 1990. Kantorovich LV, Akilov GP Functional analysis. St. Petersburg, Nevsky Dialect, BHV-Petersburg, 2002. Kirillov AA, Gvishiani AD Theorems and problems of functional analysis. M., Science, 1979. Antonevich AB, Knyazev PN, Radyno Ya.V. Tasks and exercises on functional analysis. Minsk, Higher School, 1978. |
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Teaching Methods |
interactive methods of teaching (working in small groups (team), problem training) are 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 |
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11 |
Language |
Russian |
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12 |
Conditions (requirements), current control |
– test; – colloquium
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13 |
Appraisal Form |
credit |