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Обычная версия сайта
2019/2020

Введение в функциональный анализ

Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Лучший по критерию «Новизна полученных знаний»
Статус: Дисциплина общефакультетского пула
Когда читается: 1, 2 модуль
Язык: английский
Кредиты: 6

Course Syllabus

Abstract

Functional analysis studies infinite-dimensional vector spaces equipped with a norm (or, more generally, with a topology), operators between such spaces, and representations of algebraic structures on such spaces. The classical areas of Functional Analysis are the spectral theory of linear operators, the geometry of Banach spaces, distribution theory, operator algebra theory, etc. Among relatively new areas are noncommutative geometry (in the spirit of Connes), operator space theory (a.k.a. "quantum functional analysis"), and locally compact quantum groups. Functional analysis has numerous applications in differential equations, harmonic analysis, representation theory, geometry, topology, calculus of variations, optimization, quantum physics, etc. In this introductory course, we plan to cover the very basics of Functional Analysis (the "irreducible minimum") only.
Learning Objectives

Learning Objectives

  • Students will be introduced to the basic notions and the basic principles of Functional Analysis.
Expected Learning Outcomes

Expected Learning Outcomes

  • Prove the completeness of classical function spaces
  • Calculate the norms of linear operators
  • Apply the basic principles of Functional Analysis in concrete situations.
  • Find the spectra of linear operators by using, in particular, duality theory
Course Contents

Course Contents

  • Normed and Banach spaces, bounded linear maps.
  • Hilbert spaces.
  • The Hahn-Banach Theorem, the Open Mapping Theorem, the Uniform Boundedness Principle.
  • Compact operators. The Hilbert-Schmidt Theorem.
  • Elementary spectral theory.
  • Basic duality theory.
Assessment Elements

Assessment Elements

  • non-blocking midterm grade
    The midterm exam (oral) will be at the end of October (or at the beginning of November) and will include only the material of the 1st module.
  • non-blocking exercise sheets grade
    To get the maximum grade for the exercise sheets, you should solve 75% of all the exercises. If you solve more, you will earn bonus points. You can also earn bonus points for working actively at the exercise classes and for solving «bonus exercises» (marked as «B» in the sheets).
  • non-blocking final exam
    The oral exam will be at the end of December and will include only the material of the 2nd module.
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    0.35 * exercise sheets grade + 0.3 * final exam + 0.35 * midterm grade
Bibliography

Bibliography

Recommended Core Bibliography

  • Simon, B. (2015). Real Analysis. Providence, Rhode Island: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1347487

Recommended Additional Bibliography

  • Reed, M. (1972). Methods of Modern Mathematical Physics : Functional Analysis. Oxford: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=567963