Laurea magistrale in Matematica

Analysis

ANALYSIS

Anno accademico 2022/2023

• Analysis (Course A) (MAT0032)
  Laurea Magistrale (M.Sc.) in Stochastics and Data Science

Sommario insegnamento

• Obiettivi formativi
• Risultati dell'apprendimento attesi
• Modalità di insegnamento
• Modalità di verifica dell'apprendimento
• Programma
• Testi consigliati e bibliografia
• Strumenti didattici

Obiettivi formativi

The course introduces the participants to the theory of infinite-dimensional vector spaces and of linear operators between them, with a special focus on the concepts of normed vector spaces, completeness, compactness, and the different topologies which characterize the infinite dimensional vector spaces. Applications concern various spaces of functions and operators between them (in particular, integral and differential operators). The course presents basic tools of modern mathematical analysis which are of fundamental importance in many branches of pure and applied mathematics, in particular in probability theory, statistics, numerical analysis, partial differential equations and dynamical systems.

Risultati dell'apprendimento attesi

Students will acquire knowledge and understanding of many basic tools which are of common use in the analysis of infinite dimensional vector spaces. In particular he will learn the theory of Banach and Hilbert spaces and  their dual spaces, of linear, bounded, and compact operators, and he will know the theory of distributions (generalized functions), as well as the Fourier and Laplace transform.

Students will be able to solve simple problems and exercises related to the theory (in particular, to solve simple integral or differential equations) and he will be able to rigorously prove main results of the theory. 

Students will be able to select the appropriate method to solve problems and exercises related to the theory.

Students will properly use mathematical language to prove the theorems and solve exercises related to the theory.

Students will acquire critical thinking abilities, as well as capacities of collaborating. These skills will help the students to improve their learning capacities.

Modalità di insegnamento

The course is articulated in 72 hours of formal in‐class lecture time, and in at least 150 hours of at‐home work solving practical exercises. 

The lectures will be in presence with exceptions in accordance with the university regulations. 

Modalità di verifica dell'apprendimento

The assessment consists in a written test followed by an oral examination, after completion of the course.

The written test consists in open questions and exercises on the topics treated in class and has a duration of 180 minutes. The mark will be expressed in thirtieth; the single points (30 in total) will be distributed to the questions and exercises on the basis of their importance and length; the final score will be given by summing up the partial scores of each question and exercise. 

The oral examination is scheduled after the written test and can be given only after having passed the written test with a mark of 18 or better. The oral examination consists of questions on the written test and on the topics treated in class and listed in the examination programme (which is available to the participants on the web-site of the course).

Both written test and oral examination will result in a final mark expressed in thirtieth; the minimal mark allowed for successful assessment is 18. Otherwise, the student's performance is considered insufficient and the student has to repeat the examination (both written test and oral examination).

Both written test and oral examination must be achieved in the same examination period. 

The final grade will be a combination of the written test grades (90%) and the oral exam grade (10%).

The exams will be in presence with exceptions in accordance with the university regulations. 

Programma

• Banach spaces.
• Linear operators.
• Hilbert spaces, projections, orthonormal basis.
• Generalized Fourier series.
• Dual spaces: linear functionals, weak convergence.
• Compactness in finite dimensional spaces.
• Compact operators and applications to integral equations.
• Fundamentals of spectral theories
• Distributions (generalized functions)
• Fourier transform
• Laplace transform

• Bryan P. Rynne and Martin A. Youngson, Linear Functional Analysis, Second Edition, Springer, 2008.
• Dudley, R.M., Real Analysis and Probability, Cambridge University Press.
• Hörmander, L., The Analysis of Linear Partial Differential Operators I, Distribution Theory and Fourier Analysis, Springer, 2003.
• Royden, H.L., Real Analysis, MacMillan.
• Rudin, W., Functional Analysis, McGraw-Hill.
• Rudin, W., Real and Complex Analysis, McGraw-Hill.

Additional Lecture Notes will be made available to the students.

Orario lezioni

• Materiale didattico
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