# ELEMENTS OF ADVANCED ANALYSIS 1

7 credits during the 1st year of 9011 Mathematics (LM-40) GENOVA

OVERVIEW

Some basic topics in Mathematical Analysis are covered, with the aim to continue the study already begun in the previous courses of Mathematical Analysis I, 2 and 3.

## 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 knowlodge of Mathematicst 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 and 3, 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

- Normed and Banach Spaces, bounded linear operators.
- Hahn-Banach theorem, Banach-Steinhaus theorem, open mapping and closed graph theorems.
- Hilbert spaces, orthonormal bases, generalised Fourier series.
- Riesz representation theorems and Hilbert projection theorem.
- 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.
- Radon-Nikodym theorem.

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 - Analisi reale e complessa - Bollati Boringhieri
- 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

**Ricevimento:** By appointment

**Ricevimento:** At the end of lectures or by appointment.

Exam Board

GIOVANNI ALBERTI (President)

MATTEO SANTACESARIA

ADA ARUFFO (President Substitute)

## LESSONS

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.

LESSONS START

The class will start according to the academic calendar.

## EXAMS

Exam description

The exam consists in a written test and in 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

Date | Time | Location | Type | Notes |
---|---|---|---|---|

09/06/2021 | 09:00 | GENOVA | Scritto | |

10/06/2021 | 09:00 | GENOVA | Orale | |

12/07/2021 | 09:00 | GENOVA | Scritto | |

13/07/2021 | 09:00 | GENOVA | Orale | |

15/09/2021 | 09:00 | GENOVA | Scritto | |

15/09/2021 | 14:00 | GENOVA | Orale |