## OVERVIEW

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

The lectures are delivered in Italian.

## AIMS AND CONTENT

LEARNING OUTCOMES

Introduction of further concepts of Lebesgue's Integration Theory and some basic ones of Functional Analysis.

AIMS AND LEARNING OUTCOMES

To provide some fondamental contents in Mathematical Analysis (Functional Analysis and Measure Theory) that are considered important to get a well grounded knowledge in the basic branches of Mathematics and for the students who want to get a master's degree in Mathematics.

**Expected learning outcomes:**

The students will become acquainted with the concepts and proofs carried out in class and how they are used in practice to solve exercises; moreover they will know how to produce easy variants of demonstrations seen and construct examples on topics covered in this course.

TEACHING METHODS

During the semester some exercitations will carried out in class. Both theory and exercises are presented by the teacher in the classroom on the blackboard. Moreover some tutorial exercitations can be carried out during the semester, if some students will be interested. The teaching materials is available at the teacher's web page.

SYLLABUS/CONTENT

Normed and Banach spaces; continuous operators. Hahn Banach, uniform boundedness, open map and closed graph theorems; Hilbert spaces, Riesz and projection theorems, L^p spaces. Convergences of measurable functions..Radon-Nikodym theorem; duals of L^p and C_{infinity}; bounded variation and absolutely continuous functions.

RECOMMENDED READING/BIBLIOGRAPHY

H. Brezis - Analyse Fonctionnelle, Theorie et applications - Masson 1983.

N. Dunford, J.T. Schwartz - Linear Operators. Part I: General Theory - Interscience 1957.

W. Rudin - Real and Complex Analysis - McGraw-Hill 1970

A.E. Taylor, D.C. Lay - Introduction to Functional Analysis - Wiley and Sons 1980.

## TEACHERS AND EXAM BOARD

**Office hours:** At the end of lectures or by appointment.

## LESSONS

TEACHING METHODS

During the semester some exercitations will carried out in class. Both theory and exercises are presented by the teacher in the classroom on the blackboard. Moreover some tutorial exercitations can be carried out during the semester, if some students will be interested. The teaching materials is available at the teacher's web page.

LESSONS START

The class will start according to the academic calendar.

Class schedule

All class schedules are posted on the EasyAcademy portal.

## EXAMS

EXAM DESCRIPTION

Written and oral tests.

ASSESSMENT METHODS

The written examination consists in some exercises about the topics covered in this course. In the oral exam a discussion of the written examination is done; moreover some questions are asked on the course content.

Exam schedule

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

26/01/2018 | 10:00 | GENOVA | Scritto | |

29/01/2018 | 10:00 | GENOVA | Orale | |

16/02/2018 | 10:00 | GENOVA | Scritto | |

19/02/2018 | 10:00 | GENOVA | Orale | |

22/06/2018 | 10:00 | GENOVA | Scritto | |

26/06/2018 | 10:00 | GENOVA | Orale | |

20/07/2018 | 10:00 | GENOVA | Scritto | |

23/07/2018 | 10:00 | GENOVA | Orale | |

14/09/2018 | 10:00 | GENOVA | Scritto | |

17/09/2018 | 10:00 | GENOVA | Orale |