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

Introduction to the fundamental concepts of functional analysis.

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 1, 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
• Gerald Folland - Real Analysis: Modern Techniques and Their Applications - John Wiley & Sons 1999

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

See https://corsi.unige.it/corsi/11897/studenti-orario.

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.

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.

FURTHER INFORMATION

Compensatory and dispensatory measures Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)
• Additional time (+50% disability/invalidity)
• Additional time during oral exams to organise the answer
• Calculator (programmable and graphing calculators are not allowed)
• Conceptual Maps
• Tables and/or Forms
• Take the exam in written form
• Take the exam in oral form
• Tutor reader (for written tests only)
• Tutor-writer (for written tests only)

Your request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it