MAT-61056 Advanced Functional Analysis, 5 cr

Lisätiedot

Suitable for postgraduate studies.

Lassi Paunonen

Opetus

 Toteutuskerta Periodi Vastuuhenkilö Suoritusvaatimukset 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.

Osaamistavoitteet

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.

Sisältö

 Sisältö Ydinsisältö Täydentävä tietämys Erityistietämys 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

Oppimateriaali

 Tyyppi Nimi Tekijä ISBN URL Lisätiedot Tenttimateriaali 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

Esitietovaatimukset

 Opintojakso P/S Selite MAT-60206 Mathematical Analysis Mandatory MAT-61007 Introduction to Functional Analysis Mandatory MAT-61757 Measure and Integration Mandatory

Tietoa esitietovaatimuksista
Requires a strong background in analysis, fundamental functional analysis and measure and integration theory.

Vastaavuudet

Opintojakso ei vastaan mitään toista opintojaksoa

 Päivittäjä: Kunnari Jaana, 05.03.2019