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 |
Attendance and activity in the weekly teaching sessions (40%), submitted weekly homework problems (40%), and submitted pre-session exercise problems (20%). For details on the topics and grading, see the course Moodle page. |
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