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Обычная версия сайта
2020/2021

Введение в функциональный анализ

Лучший по критерию «Полезность курса для Вашей будущей карьеры»
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
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
  • Identify the duals of concrete Banach spaces and operators
  • Find the spectra of linear operators by using, in particular, duality theory
  • Prove the compactness or noncompactness of concrete operators
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
  • Basic duality theory
  • Elementary spectral theory
  • Compact operators. The Hilbert-Schmidt Theorem
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

  • Vladimir Kadets. (2018). A Course in Functional Analysis and Measure Theory (Vol. 1st ed. 2018). Springer.

Recommended Additional Bibliography

  • John B Conway. (1985). A Course in Functional Analysis. Springer.