00530 - Fundamentals of Advanced Analysis

Learning outcomes

At the end of the course, students should know modern advanced tools in mathematical analysis: Fourier transform, Hilbert and Banach spaces, abstract measure theory, weak derivatives. Student should be able to use these tools to attack and solve non-elementary problems in applied sciences, in particular those that can be modeled in terms of partial differential equations. Finally, they should reach learning skills and high level ok knowledge, making them able to accede to master lectures and programs.

Course contents

1) Elements of abstract measure theory. Borel measures, Radon measures. Lebesgue decomposition. Radon-Nikodym theorem.
2) L^p spaces: completeness, density of some classes of functions (simple functions, continuous functions), regularization (Friedrichs' mollifiers). 3) Banach and Hilbert spaces. Baire Theorem, Hahn-Banach, Banach-Steinhaus and closed graph Theorems. 4) Fourier transform il L^1, in the Schwarz space S, and in L^2. 5) Weak derivatives and Sobolev spaces. Lax-Milgram Theorem and Dirichlet problem for second order elliptic operators.

Readings/Bibliography

H. Brezis, Sobolev Spaces and Partial Differential Equations, Springer, New York.
W. Rudin, Analisi reale e complessa, Boringhieri, Torino

Teaching methods

Lectures at the blackboard

Assessment methods

Both written and oral exam. The written exam will consist of a short dissertation about one of the topics of the course (length: 2h).

Office hours

See the website of Bruno Franchi