ELEMENTS OF ADVANCED MATHEMATICAL ANALYSIS

Module MODULE 2

The main objective of the course is to provide students with an in-depth treatment of the fundamental concepts and results of real analysis, thereby enriching their background in mathematical analysis and providing them with useful tools for studying topics arising in other courses (harmonic analysis, ordinary or partial differential equations, etc.). This will be achieved by studying the main non-elementary properties of Lebesgue spaces, the theory of real-valued functions of one real variable that are of bounded variation or absolutely continuous, and Sobolev spaces in one dimension. These are substantial topics, closely related to familiar results from differential and integral calculus. In particular, the course pursues the following objectives:

Knowledge and understanding

The main non-elementary properties (separability, uniform convexity, etc.) of 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 address real-valued functions of one real variable that are of bounded variation or absolutely continuous, treating in detail the integration-by-parts formula and the differentiation and integration of composite functions. Finally, Sobolev spaces in one dimension will be studied, as well as the Cauchy and 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 and absolutely continuous functions, to study weak convergence in Lᵖ spaces and in Sobolev spaces in one dimension, and to handle simple boundary value problems under Carathéodory-type hypotheses.

Making judgements

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

Communication skills

During lectures, students will be constantly encouraged to participate and express their views on both theoretical topics and applications. At the end of the course, they may give a seminar on a topic agreed upon with the instructor. This is intended to develop their critical thinking and intuition, as well as to train them to communicate using mathematically correct language.

Learning skills

These skills will be fostered 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 problem solving, critical discussion of cases and examples, and 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 plays an increasingly important role in the final part, devoted to Sobolev spaces and boundary value problems. This is intended to consolidate students' ability to apply the techniques learned independently and to develop the autonomy of judgement and communication skills specified among the expected learning outcomes. The final seminar agreed upon with the instructor also provides a specific opportunity to develop 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 fundamentals of abstract measure and integration theory, Lebesgue spaces, and the main results of elementary functional analysis (Banach and Hilbert spaces, convergence criteria). Such knowledge is generally acquired in a basic course on measure and integration theory and linear functional analysis. These prerequisites are 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 topics that are often non-elementary and helps students develop the analytical skills, the ability to make connections among the various topics of the course, and the independent problem-solving skills required in the examination.

Further topics on Lᵖ spaces: separability, approximation by smooth functions, uniform convexity, duality, weak convergence, Radon-Nikodym theorem. Real-valued functions of one real variable of bounded variation: Jordan's theorem, Vitali's covering lemma, differentiation and Lebesgue's theorem. Absolutely continuous real-valued functions of one real variable: integral functions and their derivatives, determination of antiderivatives, comparison among classes of functions (Cantor set and Cantor-Lebesgue function), chain rule, change of variables in integrals, rectifiable curves. Sobolev spaces in one dimension: mollifiers, selected function 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 oral test, during which some simple exercises will also be proposed. The assessment will take into account clarity of exposition, completeness of knowledge, and the ability to make connections between different topics. Students must demonstrate sufficient knowledge of the main topics covered in the course and the ability to solve at least the simplest of the proposed exercises.

Grading criteria (EN)

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 ability to present and connect the course contents is modest; they are able to solve simple exercises.

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

28-30 cum laude: the student has acquired all the course contents and is able to present them thoroughly and connect them critically; 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 their department (https://www.cinap.unict.it/content/referenti).

1.   Discuss the separability or uniform convexity of 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 function 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.