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 of the fundamental concepts of Lebesgue's Integration Theory and 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 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

• Complements of Normed and Banach Spaces.
• Hahn-Banach theorem, Banach-Steinhaus theorem, open mapping and closed graph theorems.
• L^p spaces: Hölder and Minkowsky inequalities, Riesz-Fischer theorem, density properties.
• Convergences of measurable functions: convergence in measure, almost uniform convergence, Severini-Egoroff theorem.
• The Radon-Nikodym theorem.
• The dual of L^p.
• The dual of the space of continuous functions vanishing at infinity.

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

Exam Board

GIOVANNI ALBERTI (President)

MATTEO SANTACESARIA

ADA ARUFFO (President Substitute)

LESSONS

LESSONS START

The class will start according to the academic calendar.

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.

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.

Exam schedule

FURTHER INFORMATION

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take into account the individual learning arrangements and provide appropriate compensatory tools.