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Complementary chapters of functional analysis/ Module 1. Applied functional analysis.Module 2.Convex

ДВВ.03

Обов’язкова дисципліна для ОП

Second

2023/2024

1 Semester

6

PLO3. Gaining knowledge for the ability to evaluate existing technologies and on the basis of analysis to form requirements for the development of advanced information technologies.

Full-time form

To successfully learn the discipline “Complementary chapters of functional analysis. Module 1. Applied Functional Analysis” the student should satisfy the following requirements. They have successfully passed the courses Calculus, Functional analysis, and Linear Algebra. They know (a) fundamentals of Calculus, Functional analysis, and Linear Algebra . They can (a) solve problems of Calculus, Functional analysis, and Linear Algebra. They should be able to (a) seek information in the Internet.

Block 1. Filters and nets. Nets Filters and bases of filters. Limits and limit points of filters. Ultrafilters. Criteria of compactness. Connection between filters and nets Nopology induced by a family if sets Tikhonov topology Test work Block 2. Topological vector spaces. Fundamentals of topological vector spaces Completeness and compactness in topological vector spaces Linear operators and functionals Locally convex spaces Weak topologies General notions of duality Duality in locally convex spaces Duality in Banach spaces Krein–Milman theorem Test work Part 3. Elements of convex analysis. Convex sets and functions Separability theorems Duality of convex functions Subdifferential calculus Extreme conditions Duality of convex problems Part 4. Elements of nonlinear analysis. Theorems of Brouwer and Schauder Kakutani's theorem Browder's theorem, the Krasnoselskyi-Mann method, the Halpern method. Von Neumann's alternating method. Basic concepts of the theory of monotone operators, Minty's lemma. Variational inequalities Methods of solving variational inequalities.

1. Kadets V.M. Kurs funktsional-nogo analiza. — Kh.: KhNU im. V.N. Karazina, 2006. — 608 s. 2. Aleksandrian R.A., Mirzakhanian E.A. Obshchaia topologiia. - M.:Vysshaia shkola, 1979. - 336 s. 3. Berezanskii Iu.M., G.F.Us, Sheftel- Z.G. Funktsional-nyi analiz. - K.: Vishcha shkola, 1990. - 600 s. 4. Kantorovich L.V., Akilov G.P. Funktsional-nyi analiz. - M.: Nauka, 1984. - 752 s. 5. Kolmogorov A.N. Fomin S.V. Elementy teorii funktsii i funktsional-nogo analiza.- M: Nauka, 1981. - 544 s. 6. Kelli Dzh. Obshchaia topologiia. – M.: Nauka, 1981. 7. Engel-king R. Obshchaia topologiia. – M.: Mir, 1986.

Lectures, independent work, recommended literature processing, homework.

Intermediate assessment: The maximal number of available points is 60. Test work no. 1: RN 1.1, RN 1.2 – 30/18 points. Test work no. 2: RN 1.1, RN 1.2 – 30/18 points. Final assessment (in the form of exam): The maximal number of available points is 40. The results of study to be assessed are RN 1.1, RN 1.2, RN 2.1, and RN 3.1. The form of exam: writing. The types of assignments are 4 writing assignments (2 theoretical and 2 practical).

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