81657 - Mathematical Analysis 3

Learning outcomes

At he end of the lessons, the student knows some advanced modern mathematical analysis instruments: Hilbert and Banach spaces as well a some linear operators between these spaces, weak convergence, L^p spaces, Fourier transform in L^2,  Ascoli-Arzelà theorem. The student knows how to use this knowledge for facing and solving applied problems. He or She is habit to learn and is endowed of some specific skills for attending to the second order degree lessons.

Course contents

1) L^p spaces: completeness, density of some set of functions (simple functions, continuous functions), regularization (Friedrichs mollificators).

2) Fourier transform in L^1, in Schwarz space S and in L^2. Introduction to the temperate space distribution S' and to the distribution space D'. Basic to Laplace transform. 

3) Weak derivatives and Sobolev spaces. Lax-Milgram theorem and the Dirichlet problem for second order elliptic operators.

4) Banach spaces, Hilbert spaces. Baire's theorem. Hahn-Banach theorem, Banach-Steinhaus theorem and closed graph theorem. Weak convergrnce and minima of coercive functionals and weakly lower semicontinuous functionals.

5) Basic abstrac measure theory. Borel and Radon measures. Lebesgue decomposition, Radon-Nikodym Theorem.

 

 

Readings/Bibliography

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

Teaching methods

Classical lessons.

Assessment methods

Didactic targets:

-to be able to introduce some topics discussed during the course, proving to have deeply understood the fundamental theoretical concepts;

-to be able of solving some exercises.

The exam is subdivided in a 2 hours written exam.

In order to be admetted to the colloquium trial, the candidate has to obtain at least 10/18 in the the written exam.

The final score is determinedadding or subtracting to the written grade even 15 points decided by the commission during the colloquium.

Teaching tools

The lecture notes will be available on Virtuale.

Links to further information

https://www.unibo.it/sitoweb/fausto.ferrari

Office hours

See the website of Fausto Ferrari