Academic Year 2021-2022

FOUNDATIONS OF HIGHER ANALYSIS

Teachers

Gianluca Gorni
Duccio Papini
Unit Credits
12
Teaching Period
Annuity
Course Type

FOUNDATIONS OF MATHEMATICAL ANALYSIS - 1ST MODULE

Duccio Papini
Prerequisites. Inalienable prerequisites: the topics taught in the courses Mathematical Analysis 1 and 2
Objectives
The student is required of:

– Knowing the foundations Functional Analysis that will constitute the main topic of the couse.

– Knowing how to apply abstract results to concrete problems, for instance in dealing with solutions to partial differential equations.

Expected learning outcomes as shown in the following Dublin Descriptors.

Skills related to the disciplines:

Knowledge and understanding: the student will have to know and understand some selected topics in advanced analysis and the foundations of functional analysis.

Applying knowledge and understanding:

– the student will be able to apply the main theorems and tools from functional analysis and to self-compile rigorous mathematical proofs.

Transversal skills / soft skills:

– Making judgments: the student will have to demonstrate a good independent judgment in choosing the most appropiate techniques to solve problems arising from the theory as well as from applications.

– Communication skills: the student will have to show good communication skills, being able to invent rigorous mathematical proofs, and to formulate appropiate conjectures.

– Learning skills: the student will have to demonstrate good learning ability, and to be able to study independently.

https://www.uniud.it/it/didattica/info-didattiche/regolamento-didattico-del-corso/LM-matematica/all-B2

Contents
The course will propose some topics from advanced Analysis and Functional Analysis, that constitute an indispensable part of the knowledge of any Laureato Magistrale in Mathematics. Typical topics are, for instance, distribution theory, Banach and Hilbert spaces, spectal and semigroup theory, with applications to partial differential equations and harmonic analysis.
Texts
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. (edito in Italiano da Liguori Editore)

Functional analysis. Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991.

FOUNDATIONS OF MATHEMATICAL ANALYSIS - 2ND MODULE

Gianluca Gorni
Prerequisites. Calculus in one and several variables, as taught in the first two years of Mathematics. Basics of linear algebra. Lebesgue integral. The fundamental theorems of Banach spaces, as seen in Module I
Objectives
Basics of Functional Analysis of compact linear operators in Hilbert spaces. Introduction to Fourier transform.

https://www.uniud.it/it/didattica/info-didattiche/regolamento-didattico-del-corso/LM-matematica/all-B2

Contents
Basics of Hilbert spaces. Orthogonal projections. Compact and self-adjoint operators. Fredholm equations. Sturm-Liouville’s problem. Basic theory of Fourier transform in L^1 and L^2.
Texts
W. Rudin, Analisi Reale e Complessa. Boringhieri. Haim Brezis, Analisi funzionale, teoria e applicazioni, Liguori.

Course notes from the teacher.