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Analysis (Course B)

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Analysis (Basics)

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Academic year 2024/2025

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 - mathematical analysis
Delivery
Class Lectures
Language
English
Attendance
Optional
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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Sommario del corso

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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 several results from 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 Lebesgue spaces of integrable functions and Fourier analysis will be provided.  

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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 with peculiar focus to Lebesgue spaces of integrable functions. He/She will also acquire a basic knowledge of Fourier  analysis.

- Applying knowledge and understanding

Students will be able to solve simple problems and exercises related to the theory as well and  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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Program

The course is divided into 3 parts:
  • Abstract vector spaces
  1. Banach spaces: fundamental properties, examples;
  2. Hilbert spaces; fundamental properties, projection theorem.
  • Lebesgue spaces 
  1. Definition and properties of Lp spaces. 
  2. Inequalities
  3. Convergence of sequence, weak and strong.
  • Fourier analysis
  1. Fourier transform in L1;
  2. Fourier transform in L2;
  3.  Applications and introduction to the Laplace transform.
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Course delivery

The course is articulated in 72 hours of lectures  and in at least 150 hours of at‐home work solving practical exercises. 

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

 

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Learning assessment methods

The course grade is determined solely on the basis of a written examination. The examination (3 hours) tests 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 4 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.

Suggested readings and bibliography

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- STROOCK, D. W. Essentials of Integration Theory for Analysis, Springer, 2011.

- ROBINSON, J. C. An Introduction to Functional Analysis, Cambridge University Press, 2020.

 - 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.

- Additional Lecture Notes will be made available to the students.



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Class scheduleV

Enroll
  • Closed
    Enrollment opening date
    13/09/2022 at 00:00
    Enrollment closing date
    30/01/2023 at 00:00
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