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Regular version of the site
2021/2022

Introduction to Functional Analysis

Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Type: Optional course (faculty)
When: 1, 2 module
Open to: students of all HSE University campuses
Language: English
ECTS credits: 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 à la 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. PREREQUISITES: Calculus, linear algebra, metric spaces, the Lebesgue integral. The course is accessible to 2nd year students and higher

Learning Objectives

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

Expected Learning Outcomes

• Apply the basic principles of Functional Analysis in concrete situations
• Calculate the norms of linear operators
• Find the spectra of linear operators by using, in particular, duality theory
• Identify the duals of concrete Banach spaces and operators
• Prove the compactness or noncompactness of concrete operators
• Prove the completeness of classical function spaces

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

• 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.
• 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).
• final exam
The oral exam will be at the end of December and will include only the material of the 2nd module.

Interim Assessment

• 2021/2022 1st module
• 2021/2022 2nd module
0.35 * exercise sheets grade + 0.3 * final exam + 0.35 * midterm grade

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.