Learning outcomes of the course unit
The course it is aimed at illustrating the main results from Functional analysis and from the Lp theory.
Course contents summary
1) Topological vector spaces, locally convex spaces.
2) Linear operators in topological vector spaces
3) Normed spaces and Banach spaces
4) Operators on normed spaces
5) Hahn-Banach theorem and its consequences.
6) Banach-Steinhaus theorem and its consequences
7) Open mapping theorem and its consequences
8) Weak topologies in Banach spaces
9) Reflexive spaces
10) Hilbert spaces: main properties
11) Projections in Hilbert spaces and their applications
12) Orthonormal systems
13) Fourier series in Hilbert spaces
15) Lp spaces.
1) Lecure notes by the teacher
2) H. Brezis. Functional analysis, Sobolev spaces and partiare differential equations, Springer Verlag 2011.
3) W. Rudin, Functional analysis, McGraw-Hill,
New York 1973.
Lectures. During the lectures the basic results of the functional analysis will be analyzed and discussed. Many examples will be provided to show how and where the
abstract results can be applied to make the students understand better the relevance of what they are studying.
Assessment methods and criteria
The exam consists of two parts: a written part and an oral part. The exam is aimed at evaluating the knowledge of the abstract results seen during the course, their proofs and the skills of the students in using such results.