REAL ANALISYS

Academic Year 2024/2025 - Teacher: Biagio RICCERI

Expected Learning Outcomes

The main goal of the course is to provide the student with a more in depth treatment of the most important concepts and results within Real Analysis. In such a way, the student will enrich his/her cultural background in the field of Mathematical Analysis and will acquire useful tools to follow other courses.

In more detail, following the Dublin descriptors, the objectives are the following:

Knowledge and understanding: the student will learn to work with the most typical concepts and techniques of Real Analysis.

Applying knowledge and understanding: the student will be guided in the ability to realize applications of the general results gradually established.

Making judgements: the student will be stimulated to study autonomously some results not developed during lessons.

Communication skills: the student will learn to expose in a clear, rigorous and concise manner.

Learning skills: the student will be able to face exercices and found proofs of simple results.

Course Structure

The course will be performed through frontal lessons. If necessary, the telematic way will be adopted. Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus. Students with disabilities and/or DSA are invited to plan, with the teacher, possible compensatory measures based on specific needs. They can also contact the referent teacher CInAP of the DMI.

Required Prerequisites

The contents of the courses of Mathematical Analysis I and II and of Topology.

Attendance of Lessons

The partecipation in the lecture classes is strongly recommended.

Detailed Course Content

Elements of Functional Analysis: linear functionals; Hahn-Banach theorem; normed spaces; continuous linear operators; weak topology; Hilbert spaces. Uniform convexity of L^p. Essentially bounded functions. Representation of continuous linear functionals in L^p. Compactness criteria in L^p. Theorem of Radon-Nikodym. Covering theorem of Vitali. Functions with bounded variation. Absolutely continuous functions. Holder continuous functions. Cantor singular function. Carathéodory’s functions. Theorem of Scorza-Dragoni. Generalized solutions for the Cauchy problem under Carathéodory’s assumptions.

Textbook Information

1. E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, 1965.

Some teacher's notes will be published on the Studium page of the course.

Course Planning

	Subjects	Text References
1	Elements of Functional Analysis (12 hours)	1, teacher's notes
2	L^p spaces (10 hours)	1, teacher's notes
3	Theorem of Radon-Nikodym, functions with bounded variation and absolutely continuous functions (20 hours)	1, teacher's notes
4	Carathéodory's functions and generalized solutions for the Cauchy problem (5 hours)	1, teacher's notes

Learning Assessment

Learning Assessment Procedures

The examen consists in an oral examination in which the student will be required to expose some definitions and some theorems (statement and proof). The assessment of learning could also be carried out by electronic means provided special circumstances should be required that. Normally, the following criteria will be applied in determing the vote of the single parts (P.C and final) of the examination:

not approved: the student has not acquired the basic concepts and is not able to solve exercises.

18-23: the student shows a minimal mastery of the basic concepts, his/her exposure and linking skills are modest, he/she is able to solve simple exercises.

24-27: the student shows a good mastery of the basic concepts, his/her exposure and linking skills are good, he/she solves exercises with a few mistakes.

28-30 cum laude: the student has acquired all the course contents and is able to expose and connect them in a complete and critic way, he/she solves exercices completely and without mistakes.

Examples of frequently asked questions and / or exercises

Theorem of Hahn-Banach

Compactness criteria in L^p spaces

Representation of continuous linear functionals in L^p spaces

Theorem of Radon-Nikodym

Covering theorem of Vitali

Almost everywhere differentiability of functions with bounded variation

Fundamental formula of the integral calculus for absolutely continuous functions

Cantor singular function

Existence theorem for the Cauchy problem under Carathéodory's assumptions

Degree Course in

Mathematics