Real Analisys

Knowledge and understanding

The main non-elementary properties (separability, uniform convexity, etc.) of the Lᵖ spaces will first be studied, with particular attention to the case p=2 and to the concept of weak convergence. The course will then move on to bounded variation real functions of one real variable and absolutely continuous functions, treating in detail the integration-by-parts formula and the differentiation and integration of composite functions. Finally, Sobolev spaces in dimension one will be studied, as well as the Cauchy and the Dirichlet problems for an ordinary differential equation under Carathéodory-type hypotheses.

Applying knowledge and understanding

Students will learn to work with functions of bounded variation or absolutely continuous, will be able to study weak convergence in Lᵖ spaces and in Sobolev spaces in dimension one, and to handle simple boundary value problems under Carathéodory-type hypotheses.

 Making judgements

By the end of the course, students will be able to judge which fundamental concepts and results of differential or integral calculus, as well as of differential equations, extend naturally to real analysis and which instead require a different treatment.

Communication skills

During the lectures, students will be constantly encouraged to take part, expressing their point of view on both theoretical topics and applications. At the end of the course, they may give a seminar on a topic agreed with the instructor. This aims to develop their critical sense and intuition, as well as to accustom them to communicating with mathematically correct language.

Learning skills

These skills will be stimulated and periodically assessed through classroom exercises and simple theoretical questions to be developed individually.

The course includes lectures (Direct Teaching), devoted to the systematic presentation of theoretical content, together with classroom tutorials (Interactive Teaching), devoted to guided exercise solving, critical discussion of cases and examples, and moments of ongoing formative assessment.

Direct Teaching prevails in the initial part of the course, devoted to acquiring the fundamental concepts and results (Lᵖ spaces, functions of bounded variation and absolutely continuous functions), while Interactive Teaching takes on an increasing weight in the final part, devoted to Sobolev spaces and boundary value problems, so as to consolidate students' ability to autonomously apply the techniques learned and to develop the judgement autonomy and communication skills required among the expected learning outcomes. The final seminar agreed with the instructor also represents a specific moment dedicated to developing communication skills.

If the course is delivered in blended or remote mode, appropriate adjustments may be made to the above, in order to ensure consistency with the syllabus.

Students must have a solid knowledge of the basic contents of abstract measure and integration theory, of Lebesgue spaces, and of the main results of linear functional analysis (Banach and Hilbert spaces, convergence criteria). Such knowledge is generally provided in a basic course on measure and integration theory and linear functional analysis. It is to be considered indispensable for a full understanding of the topics covered in the course.

Attendance is not mandatory. It is, however, strongly recommended, since active participation in lectures and tutorials significantly facilitates the understanding of often non-elementary topics and makes it easier to develop the analytical skills, the ability to connect the various topics of the course, and the independent exercise-solving skills required for the examination.

Further topics on Lᵖ spaces: separability, approximation by regular functions, uniform convexity, duality, weak convergence, Radon-Nikodym theorem. Bounded variation real functions of one real variable, Jordan's theorem, Vitali's covering lemma, differentiation and Lebesgue's theorem. Absolutely continuous real functions of one real variable, integral functions and their derivatives, search for antiderivatives, comparison between classes of functions (Cantor set and Cantor-Lebesgue function), chain rule, change of variables in the integral, rectifiable curves. Sobolev spaces in dimension one, mollifiers, some functional spaces, Poincaré inequality. Carathéodory functions, Cauchy and Dirichlet problems for an ordinary differential equation under Carathéodory-type hypotheses.

1.   G. Leoni, A first course in Sobolev spaces, AMS, Providence, 2017.

2.   C. Pucci, Istituzioni di Analisi Superiore, Unione Matematica Italiana, Bologna, 2013.

3.   A. Tesei, Istituzioni di Analisi Superiore, Bollati Boringhieri, Torino, 1997.

4.   R.L. Wheeden - A. Zygmund, Measure and Integral. An Introduction to Real Analysis (Second Edition), CRC Press, Boca Raton, 2015.

5.   C.D. Aliprantis - O. Burkinshaw, Problems in Real Analysis, Academic Press, San Diego, 1999.

6.   M. Muratori - F. Punzo - N. Soave, Esercizi svolti di analisi reale e funzionale, Società Editrice Esculapio, Bologna, 2021.

The examination consists of a single test. During the oral interview, some simple exercises will also be proposed. In the assessment, account will be taken of clarity of exposition, completeness of knowledge, and the ability to connect different topics. Students must demonstrate that they have acquired sufficient knowledge of the main topics covered during the course and are able to solve at least the simplest of the proposed exercises.

Grading criteria

Not passed: the student has not acquired the basic concepts and is unable to solve the exercises.

18-23: the student demonstrates minimal mastery of the basic concepts; their exposition and ability to connect topics are modest; they manage to solve simple exercises.

24-27: the student demonstrates good mastery of the course contents; their exposition and ability to connect topics are good; they solve the exercises with few errors.

28-30 cum laude: the student has acquired all the course contents and is able to expound them fully and connect them with critical insight; they solve the exercises completely and without errors.

Learning assessment may also be carried out on-line, should the conditions require it.

To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l'integrazione Attiva e Partecipata — Servizi per le Disabilità e/o i DSA) referring teacher within DMI (https://www.cinap.unict.it/content/referenti).

1.   Discuss the separability or uniform convexity of the Lᵖ spaces. Study weak convergence in Lᵖ. State and prove the Radon-Nikodym theorem.

2.   Prove the Vitali covering theorem. Prove that a function of bounded variation is differentiable almost everywhere.

3.   Illustrate the Cantor-Lebesgue function. Study the inclusions among the functional spaces C⁰([a,b]), Lip([a,b]), BV([a,b]), AC([a,b]), C¹([a,b]).

4.   Define the Sobolev space W¹,ᵖ(a,b) and prove its completeness.

5.   Prove the Poincaré inequality in dimension one and illustrate an application.

6.   Given a Cauchy or Dirichlet problem for an ordinary differential equation under Carathéodory-type hypotheses, discuss existence and uniqueness of the solution.