5 semester

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Title

FUNCTIONAL ANALYSIS

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Course

3, Mathematics (research and development)

3

Semester of training

5

4

Credits

2

5

Lecturer

Yablonskaya Anna Gennadevna

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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;

 

Requirements

Algebra and number theory,

Discrete Math,

Analytic geometry,

Mathematical analysis,

Differential equations,

Theory of functions of a complex variable

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Contents of the discipline

Theme 5 Hilbert spaces. Definition of a scalar product. The Cauchy-Bunyakovskii inequality. Hilbert spaces. The projection theorem. The Fourier series theorem.

Theme 6 Linear equations in Banach spaces. Invertible operators. Theorems on invertibility. Banach’s theorem on the inverse operator. The spectrum and the resolution of a linear bounded operator.

Topic 7 Conjugate spaces and adjoint operators. Linear bounded functionals. The Hahn-Banach Theorem. General form of a linear bounded functional in Hilbert and some other specific spaces. The conjugate space. The adjoint operator and its properties. Theorem on the conditions for the solvability of the linear equation

 Theme 8 Equations with compact operators. The Fredholm alternative for equations with operators of finite rank. Compact operators. Compactness of Integral Operators in Specific Spaces. Criterion for the finite-dimensionality of a normed space. The Riesz-Schauder theory for equations with compact operators in Hilbert space. The Fredholm alternative for integral equations.

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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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Language

Russian

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Conditions (requirements), current control

– test;

– colloquium

The score on the exam is set taking into account: 30% – work in laboratory and practical classes, 70% – oral examination.

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Appraisal Form

 exam