Programme

The main theorems of Functional Analysis.

Core Documentation

H. Brezis, Functional Analysis.

Type of delivery of the course

Many lessons.

Type of evaluation

Written and oral examination.

Programme

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytic and geometric form, consequences.
First and second category spaces, Baire theorem, Banach-Steinhaus theorem, open map and closed graph, applications.
Weak, closed and convex topologies, Banach-Alaoglu Theorem, separability, reflexivity and uniform convexity.
Sobolev spaces in one dimension, embedding theorems, Poincaré inequality, application to variational problems.
Spectral theory, Fredholm alternative, spectral theorem for compact and self-adjoint operators, application to variational problems.

Core Documentation

H. Brezis - Analisi Funzionale - Liguori (1986);
H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer (2010);
W. Rudin - Functional Analysis - McGraw-Hill (1991);

Type of delivery of the course

In person lectures

Type of evaluation

the test consists in carrying out exercises and presenting topics discussed in class

Programme

The main theorems of Functional Analysis.

Core Documentation

H. Brezis, Functional Analysis.

Type of delivery of the course

Many lessons.

Type of evaluation

Written and oral examination.

Programme

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytic and geometric form, consequences.
First and second category spaces, Baire theorem, Banach-Steinhaus theorem, open map and closed graph, applications.
Weak, closed and convex topologies, Banach-Alaoglu Theorem, separability, reflexivity and uniform convexity.
Sobolev spaces in one dimension, embedding theorems, Poincaré inequality, application to variational problems.
Spectral theory, Fredholm alternative, spectral theorem for compact and self-adjoint operators, application to variational problems.

Core Documentation

H. Brezis - Analisi Funzionale - Liguori (1986);
H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer (2010);
W. Rudin - Functional Analysis - McGraw-Hill (1991);

Type of delivery of the course

In person lectures

Type of evaluation

the test consists in carrying out exercises and presenting topics discussed in class

Programme

The main theorems of Functional Analysis.

Core Documentation

H. Brezis, Functional Analysis.

Type of delivery of the course

Many lessons.

Type of evaluation

Written and oral examination.

Programme

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytic and geometric form, consequences.
First and second category spaces, Baire theorem, Banach-Steinhaus theorem, open map and closed graph, applications.
Weak, closed and convex topologies, Banach-Alaoglu Theorem, separability, reflexivity and uniform convexity.
Sobolev spaces in one dimension, embedding theorems, Poincaré inequality, application to variational problems.
Spectral theory, Fredholm alternative, spectral theorem for compact and self-adjoint operators, application to variational problems.

Core Documentation

H. Brezis - Analisi Funzionale - Liguori (1986);
H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer (2010);
W. Rudin - Functional Analysis - McGraw-Hill (1991);

Type of delivery of the course

In person lectures

Type of evaluation

the test consists in carrying out exercises and presenting topics discussed in class