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1. |
Title |
FUNCTIONAL ANALYSIS |
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2. |
Course |
3, Mathematics (economic activity) |
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3. |
Semester of training |
5 |
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4. |
Credits |
4 |
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5. |
Lecturer |
Antonevich Anatoly Borisovich |
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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 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 |
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7. |
Requirements |
Theory of Probability, Mathematical Statistics, Equations of Mathematical Physics, Optimization Methods, Extremal Problems and Calculus of Variations, Numerical Methods |
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8. |
Contents of the discipline |
Theme 5. Normalized spaces. Vector spaces, normed spaces. Continuity of addition and multiplication by a number. Banach spaces. Replenishment of normed spaces. Criterion for the finite-dimensionality of a normed space. Theme 6. Hilbert spaces. Definition of a scalar product. The Cauchy-Bunyakovskii inequality. Hilbert spaces. The projection theorem. The Fourier series theorem. A criterion for the existence of a countable orthonormal basis. Isomorphism of separable Hilbert spaces. Theme 7. Linear operators in normed spaces. The connection between boundedness and continuity for linear operators. The norm of the operator. The space of linear bounded operators. Different types of convergence of linear bounded operators. The Banach-Steinhaus theorem. Inverse operators. Banach’s theorem on the inverse operator. Invertibility of an operator close to the identity. Openness of the set of invertible operators. Banach’s theorem on the inverse operator. Theme 8. Linear continuous functionals. The Hahn-Banach Extension of linear continuous functional. General form of continuous linear functionals in concrete spaces. The conjugate space. The adjoint operator and its properties. Theorem on the closure of the image of a bounded linear operator. Theme 9. Compact operators. Definitions and properties. Compactness of Integral Operators in Specific Spaces. Equations with compact operators. The Riesz-Schauder theory for equations with compact operators in Hilbert space. The Fredholm alternative for integral operators. |
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9. |
References |
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. |
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10. |
Teaching Methods |
Lectures, laboratory practice, SSI (students scientific investigation) |
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11. |
Language |
Russian |
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12. |
Conditions (requirements), current control |
– test; – colloquium The assessment on the exam is set taking into account: 30% – work in laboratory and practical classes, 70% – oral examination. |
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13. |
Appraisal Form |
examination |