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Mathematical Analysis Complements

Academic Year 2023/2024 - Teacher: Biagio RICCERI

Expected Learning Outcomes

The main objective of the course is to show how the contents of Mathematical Analysis I and II, learned by the student in an Euclidean setting, can be extended to the general framework of Banach spaces. Particular emphasis will be put on some meaningful applications of such extensions.

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

Knowledge and understanding: the student will learn to use in the setting of Banach spaces the methods learned in the courses of Mathematical Analysis I and II.

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. Learning assessment may also be carried out online, should the conditions require it. 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 first two courses of Analysis.

Attendance of Lessons

The partecipation in the lecture classes is strongly recommended.

Detailed Course Content

Differential calculus in Banach spaces. Basic notions on Banach spaces. Continuous linear operators between Banach spaces. Differentiable operators. Lagrange theorem and its applications. Diffeomorphisms of class C^1. Local inversion theorem. Implicit function theorem. Higher order derivatives. Taylor's formula. Local extrema of real-valued functions defined on a Banach space. Necessary conditions and sufficient conditions of first and second order.

Integral calculus for functions taking their values in a Banach space. Riemann integrable functions. Riemann integral. Strongly measurable functions. Lusin's theorem. Bochner integrable functions. Bochner integral. Mean theorem for integrals. Sequences of Bochner integrable functions. Dominated convergence theorem.

Ordinary differential equations in Banach spaces. Cauchy problem. Peano existence theorem. Godunov non-existence theorem. Existence and uniqueness theorem. Linear differential equations. Applications to systems of infinitely many ordinary differential equations. Applications to some classes of partial differential equations.

Textbook Information

1. H. Cartan, Differential calculus on normed spaces: a course in Analysis, 2017.

2.  E. Hille - R. S. Phillips, Functional analysis and semi-groups, American Mathematical Society, 1957.

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


Course Planning

 SubjectsText References
1Differential calculus in Banach sopaces (16 hours)1, appunti
2Integral calculus for vector-valued functions (16 hours)2, appunti
3Ordinary differential equations in Banach spaces (15 hours)1, appunti

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

Differentiable operators

Bochner integrable functions

Systems of infinitely many ordinary differential equations