The course will cover the fundamentals of Functional Analysis and will include the following topics:
I. Abstract theory of Banach spaces and applications to Lebesgue/Sobolev spaces.
1) The Hahn-Bach theorem and connections between convexity and continuity
2) Topology and duality of Banach spaces.
3) Lebesgue spaces: duality, compactness, and density results.
4) Sobolev spaces, Sobolev embedding, and Poincaré inequalities.
II. The theory of unbounded operators and applications to Partial Differential Equations.
1) The uniform boundedness principle and an introduction to unbounded operators theory.
2) Existence of solutions of PDEs using the Hahn-Banach theorem and uniform boundedness principle.
3) Compact operators and the Fredholm alternative.
4) Hilbert spaces: Lax-Milgram Theorem and applications to elliptic PDEs. Introduction to Spectral Theory (time-permitting).
Time-permitting, we will also discuss semi-groups of evolution and the Hille-Yosida theory. The course content will be essential to all students interested in PDEs and, more generally, Analysis and Applied Mathematics. We will discuss many illustrating examples to articulate the important connection between abstract theory and concrete applications. The grade will consist of a midterm oral presentation (10-15 mn) on a topic related to Functional Analysis of the student’s choice (suggestions of topics will be provided) and a take-home final exam.
Grade: Midterm oral presentation (40%) and take-home final written Exam (60%).
Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haïm Brézis
Prior exposure to measure theory and elementary topology (e.g. in 501/502) is essential.