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Department of Mathematics "Giuseppe Peano"

# Laurea Magistrale (M.Sc.) in Stochastics and Data Science

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## Analysis (Basics)

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Course ID
MAT0033
Teacher
Bertrand Lods
Year
1st year
Teaching period
First semester
Type
D.M. 270 TAF B - Distinctive
Credits/Recognition
9
Course disciplinary sector (SSD)
MAT/05 - analisi matematica
Delivery
Formal authority
Language
English
Attendance
Mandatory
Type of examination
Written
Prerequisites
A good knowledge of basic calculus and real analysis. In particular:

Functions of several variables, differential calculus;
Linear algebra: matrices, determinants, diagonalization
Series of numbers and series of functions;
Integral calculus for functions of several variables.

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## Course objectives

This course is a  9-credit course aimed at introducing and developing many of the mathematical tools necessary in many fields of Probablity, statistics and applied mathematics. It introduces in particular the theory of infinite-dimensional  vector spaces  with a special focus on the concepts of normed vector spaces, completeness, compactness, and other characteristic properties of infinite dimensional  vector spaces. Concrete applications to spaces of functions will be provided. Application of this theory to the study of ordinary differential equation and Fourier analysis will be also given.

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## Results of learning outcomes

#### - Knowledge and understanding

The student will acquire knowledge and understanding of many basic tools which are of common use in the analysis of both finite and  infinite dimensional vector spaces. In particular,  the student will learn the importance of the notion of completeness and compactness in theory of Banach and Hilbert spaces, the basic properties of linear bounded operators in Banach spaces. He will also acquire a basic knowledge of the theory of ordinary differential equations and Fourier transform.

#### - Applying knowledge and understanding

Students will be able to solve simple problems and exercises related to the theory as well as and he will be able to rigorously prove several main results of the theory.

#### - Making judgements

The student will be able to select the appropriate  method to solve  problems and exercises related to the theoretical notions introduced in class.

#### - Communication skills

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

#### - Learning skills

The students will acquire critical thinking abilities and will be able to comunicate the results of their findings. These skills will help the students to improve their learning capacities.

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## Course delivery

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.

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## Learning assessment methods

The course grade is determined solely on the basis of a written examination. The examination (2 hours and 45 minutes) test the student's ability to do the following:

1. Present briefly the main ideas, concepts and results developed in the course, also explaining intuitively the meaning and scope of the definitions and the arguments behind the validity of the result. Students will be required to know the definitions, the statements of the theorems, the idea behing the proofs and their applications.

2. Use effectively the concepts and the result to answer questions pertaining to functional analysis.

The above is accomplished by asking the student to answer 5‐6 questions. Each of the questions has an essay part, and some of the questions also have a more practical ("exercise ") part.

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## Support activities

The course includes exercises classes; extra exercises are suggested as homework.

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## Program

The course is divided into 3 parts:

•  Abstract vector spaces
1. Topological metric spaces, compactness;
2.  Metric spaces: properties, continuity of functions;
3. Banach spaces: fundamental properties, examples;
4. Hilbert spaces; fundamental properties, projection theorem.
5.  L_p spaces.
6. The space of continuous functions, Ascoli-Arzela Theorem.
•  Ordinary Differential Equations
1.  Cauchy Lipschitz theory;
2.  Classical methods of integration of ODE's (linear equations, Laplace transform, separation of variables).
•  Fourier Analysis.
1.  Fourier series, Fourier transform;
2.  Applications

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KHURI, A.I. Advanced calculus with applications in statistics. Wiley Series in Probability and Statistics.

- ROYDEN, H.L. Real Analysis. MacMillan.

- RYNNE, B. P. and Martin A. YOUNGSON, M.A., Linear Functional Analysis, Second Edition, Springer 2008

- DUDLEY, R. M., Real Analysis and Probability, Cambridge University Press.