OVERVIEW

Some basic topics in Functional Analysis are covered, with the aim to continue the study already begun in the previous course Mathematical Analysis 4.

AIMS AND CONTENT

LEARNING OUTCOMES

The main objective of this course is to provide students with the basics of functional analysis, with a particular focus on function spaces and, more generally, on Banach spaces and operators between them.

AIMS AND LEARNING OUTCOMES

Aims

The aim of this course is to teach some classical topics in Mathematical Analysis (Functional Analysis and Measure Theory), which are considered fundamental for a basic knowledge of Mathematics and for the students who plan to continue their studies with a Master's degree in Mathematics.

Expected learning outcomes

At the end of the course, the student will have to know the theoretical concepts introduced in the lectures, construct and discuss examples related to each of them (in such a way to better understand the abstract concepts), write/reconstruct the proofs seen in the lectures or easy variants of those and solve problems on the topics of the course.

PREREQUISITES

Mathematical Analysis I, 2, 3 and 4, Linear Algebra and Analitic Geometry, Geometry 1.

TEACHING METHODS

The course consists of frontal lectures carried out by the teacher where the theory is explained and where basic examples are discussed (four hours per week). These are integrated with problem lectures (one hour per week). The teaching material, including problem sheets and old exam scripts, is available in aulaweb.

SYLLABUS/CONTENT

1. Complements of Normed and Banach Spaces.
2. Hahn-Banach theorem, Banach-Steinhaus theorem, open mapping and closed graph theorems.
3. L^p spaces: Hölder and Minkowsky inequalities, Riesz-Fischer theorem, density properties.
4. The Radon-Nikodym theorem.
5. The dual of L^p.
6. Riesz-Markov-Kakutani representation theorem.
7. Riesz-Markov representation theore on the dual of the space of continuous functions vanishing at infinity.

The topics 6. and 7. are only for those students who take the course with 7 credits.

RECOMMENDED READING/BIBLIOGRAPHY

• M. Reed, B. Simon - Functional Analysis - Academic Press 1981
• B. Simon - Real Analysis, A Comprehensive Course in Analysis, Part 1 - AMS 2015
• H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer 2011
• N. Dunford, J.T. Schwartz - Linear Operators. Part I: General Theory - Interscience 1957
• W. Rudin - Real and complex analysis - McGraw-Hill Education
• A.E. Taylor, D.C. Lay - Introduction to Functional Analysis - Wiley and Sons 1980
• C.M. Marle - Mesures et Probabilités - Hermann 1974

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

23 february 2026

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of a written test and of an oral test. Only the students who pass the written test may do the oral exam. The final grade will take into accout the grade of the written test and the evaluation of the oral test.

Students with disabilities or with specific learning disorders (SLD) are reminded that in order to request exam accommodations, it is first necessary to upload the relevant certification on the University’s online services portal at servizionline.unige.it, in the “Students” section. The documentation will be verified by the University’s Office for the Inclusion of Students with Disabilities and SLD.

Afterwards, well in advance (at least 7 days) before the exam date, students must send an email to the instructor responsible for the exam, copying (cc) both the School Contact Person for the Inclusion of Students with Disabilities and SLD (sergio.didomizio@unige.it) and the aforementioned Office. The email must include the following details:

• The course title

• The date of the exam session

• The student’s last name, first name, and student ID number

• The compensatory tools and dispensatory measures deemed necessary and requested

The Contact Person will confirm to the instructor that the student is entitled to request accommodations during the exam and that such accommodations must be agreed upon with the instructor. The instructor will reply, indicating whether it is possible to implement the requested accommodations.

Requests must be submitted at least 7 days before the exam session in order to allow the instructor sufficient time to assess them. In particular, if a student intends to use concept maps during the exam (which must be significantly more concise than those used for studying), failure to submit them within the required timeframe may result in insufficient time to make any necessary adjustments.

For more information regarding the request for services and accommodations, please refer to the document (in Italian): Linee guida per la richiesta di servizi, di strumenti compensativi e/o di misure dispensative e di ausili specifici.

ASSESSMENT METHODS

In the written test the students need to solve some problems, related to the topics of the course. This allows to evaluate the ability of the students to solve problems and to apply the theoretical results in concrete situations.

During the oral exam, the written test, the theoretical results and problems are discussed. This allows to test the knowledge of the theory of the students and their abilities to put it into practice.