MAT-61056 Advanced Functional Analysis, 5 cr

Additional information

Suitable for postgraduate studies.

Person responsible

Lassi Paunonen

Lessons

Implementation Period Person responsible Requirements
MAT-61056 2019-01 4 Lassi Paunonen
Completed final exam and weekly exercises. For details on the topics and grading, see the POP page for the course.

Learning Outcomes

The student is introduced to advanced techniques in linear operator theory and fundamental results in infinite-dimensional normed linear spaces. The main focus is in the theory of closed operators. The results and techniques are linked to the analysis of linear partial differential equations. In addition, the student will learn advanced proof techniques in functional analysis. After the course the student will be able to employ abstract operator techniques and the theory of Sobolev spaces in studying the existence and regularity of solutions of linear elliptic partial differential equations.

Content

Content Core content Complementary knowledge Specialist knowledge
1. Closed operators on Banach spaces, definition and characterizations  Basic techniques, verifying closedness in example cases  Identifying and utilizing closedness of an operator as a tool for analysis 
2. Baire Category Theorem, Closed-Graph Theorem, Open Mapping Theorem, Uniform Boundedness Principle  Understanding the outlines of the proofs and the main applications of the results  Understanding the technical deails of the proofs and identifying the utilized proof techniques 
3. Operator representation of partial differential equations and the fundamental nature of differential operators  Ability to cast simple PDE examples in the operator theoretic framework  Ability to cast complicated PDE examples in the operator theoretic framework 
4. Theory of Sobolev spaces, definitions and fundamental meaning  Identifying the role of Sobolev spaces in the analysis of PDEs  Ability to independently utilize Sobolev spaces in the analysis of PDEs 
5. Fredholm theory and regularity for PDEs, fundamental concepts  Understanding of the main role of Fredholm theory in the analysis of PDEs  Analysis of spectrum and regularity of solutions of PDEs 

Study material

Type Name Author ISBN URL Additional information Examination material
Book   Functional Analysis, Sobolev Spaces, and Partial Differential Equations   Haim Brezis       Available through TUNI library.   Yes   
Book   Partial Differential Equations: Second Edition   Lawrence C. Evans       Supplementary material   No   
Book   Real and Complex Analysis   Walter Rudin       Supplementary material.   No   

Prerequisites

Course Mandatory/Advisable Description
MAT-60206 Mathematical Analysis Mandatory    
MAT-61007 Introduction to Functional Analysis Mandatory    
MAT-61757 Measure and Integration Mandatory    

Additional information about prerequisites
Requires a strong background in analysis, fundamental functional analysis and measure and integration theory.

Correspondence of content

There is no equivalence with any other courses

Updated by: Kunnari Jaana, 05.03.2019