Advanced Functional Analysis, 5 op

Ydinsisältö

• Closed operators on Banach spaces, definition and characterizations
• Baire Category Theorem, Closed-Graph Theorem, Open Mapping Theorem, Uniform Boundedness Principle
• Operator representation of partial differential equations and the fundamental nature of differential operators
• Theory of Sobolev spaces, definitions and fundamental meaning
• Fredholm theory and regularity for PDEs, fundamental concepts

Täydentävä tietämys

• Basic techniques, verifying closedness in example cases
• Understanding the outlines of the proofs and the main applications of the results
• Ability to cast simple PDE examples in the operator theoretic framework
• Identifying the role of Sobolev spaces in the analysis of PDEs
• Understanding of the main role of Fredholm theory in the analysis of PDEs

Erityistietämys

• Identifying and utilizing closedness of an operator as a tool for analysis
• Understanding the technical deails of the proofs and identifying the utilized proof techniques
• Ability to cast complicated PDE examples in the operator theoretic framework
• Ability to independently utilize Sobolev spaces in the analysis of PDEs
• Analysis of spectrum and regularity of solutions of PDEs