REAL ANALISYS

The main objective of the course is to provide the student with an in-depth treatment of the basic concepts and results inherent in real analysis, in order to enrich their cultural background in the field of mathematical analysis and to provide a useful tool for the study of issues coming from other teachings (harmonic analysis, ordinary differential equations or partial differential equations, etc.). This will be achieved by initially examining the theory of real functions of a real variable with limited variation and absolutely continuous. This is a rich chapter, which has a close relationship with known facts of differential and integral calculus.

In particular, the course has the following objectives:

 Knowledge and understanding: the main topics inherent to the real functions of a real variable with limited variation and absolutely continuous will be studied first, also with the aim of deepening and unifying some notions and methodologies learned in previous courses of mathematical analysis. After the necessary reminders on Banach and Hilbert spaces, L^p spaces will be examined, with particular attention to the case p=2 and the concept of weak convergence, as well as Sobolev spaces in dimension one. Finally, the Cauchy problem for an ordinary differential equation under the Carathéodory hypothesis will be studied.

Applying knowledge and understanding: the student will learn to operate with functions with bounded variation or absolutely continuous, will be able to study weak convergence in L^p spaces and in Sobolev spaces in dimension one, as well as deal with simple boundary value problems under the Carathéodory hypothesis.

Making judgements: at the end of the course the student will be able to judge which of the fundamental concepts and results of differential or integral calculus extend naturally to real analysis.

Communication skills: during the lessons the students will be constantly invited to intervene, expressing their point of view, both on theoretical topics and on applications. This is aimed at developing their critical sense and intuition, as well as getting them used to communicating with mathematically correct language.

Learning skills: they will be stimulated and periodically verified with classroom exercises and simple theoretical questions to be developed individually.

Frontal lessons and classroom exercises. 

The learning assessment includes a single test, with the completion of some simple exercises. It can also be done online, if conditions require it. 

PLEASE NOTE: Information for students with disabilities and/or DSA 

To ensure equal opportunities and in compliance with current laws, interested students can request a personal interview in order to plan any compensatory and/or dispensatory measures, based on the educational objectives and specific needs.

It is also possible to contact the CInAP (Active and Participatory Integration Center - Services for Disabilities and/or DSA) contact teacher of our department, Prof. Daniele, or the President of the Degree Course.

Vitali's covering lemma. Cantor set. Real functions of one real variable with bounded variation and absolutely continuous. Jordan and Lebesgue theorems. Cantor-Lebesgue function. Complements on continuous functions: approximation by polynomials; Ascoli-Arzelà theorem. Weak topology in a Banach space. L^p spaces: main p Vitali's covering lemma. Cantor set. Real functions of one real variable with bounded variation and absolutely continuous. Jordan and Lebesgue theorems. Cantor-Lebesgue function. Complements on continuous functions: approximation by polynomials; Ascoli-Arzelà theorem. Weak topology in a Banach space. L^p spaces: main properties; weak convergence; approximation by regular functions. Sobolev spaces in one dimension. Carathéodory functions. Study of the Cauchy problem for an ordinary differential equation under the Carathéodory hypotheses.

1. C. Miranda, Istituzioni di Analisi Funzionale Lineare, Unione Matematica Italiana, Bologna, 1978.
2. C. Pucci, Istituzioni di Analisi Superiore, Unione Matematica Italiana, Bologna, 2013.
3. R.L. Wheeden - A. Zygmund, Measure and Integral. An Introduction to Real Analysis (Second Edition), CRC Press, Boca Raton, 2015.
4. M. Muratori - F. Punzo - N. Soave, Esercizi svolti di analisi reale e funzionale, Società Editrice Esculapio, Bologna, 2021.

Before the oral exam, a simple written test will be carried out. The evaluation will take into account the clarity of the presentation, the completeness of the knowledge and the ability to connect different topics. The student must demonstrate that he/she has acquired sufficient knowledge of the main topics covered during the course and that he/she is able to complete at least the simplest of the assigned exercises. The following criteria will normally be used to assign the grade: 

not approved: the student has not acquired the basic concepts and is unable to complete the exercises. 

18-23: the student demonstrates a minimal mastery of the basic concepts, his/her ability to present and connect the contents is modest, he/she is able to solve simple exercises. 

24-27: the student demonstrates a good mastery of the course contents, his/her ability to present and connect the contents is good, he/she solves the exercises with few errors. 

28-30 cum laude: the student has acquired all the contents of the course and is able to present them fully and connect them with a critical spirit; solves the exercises completely and without errors. The learning assessment may also be carried out electronically, if conditions require it.