Analysis (Course B)
Academic year 2020/2021
- Course ID
- Bertrand Lods
- 1st year
- Teaching period
- First semester
- D.M. 270 TAF B - Distinctive
- Course disciplinary sector (SSD)
- MAT/05 - analisi matematica
- Formal authority
- Type of examination
- 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.
Sommario del corso
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.
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 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 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 skillsThe 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.
ProgramThe course is divided into 3 parts:
- Abstract vector spaces
- Banach spaces: fundamental properties, examples;
- Hilbert spaces; fundamental properties, projection theorem.
- The space of continuous functions, Ascoli-Arzela Theorem.
- Lebesgue spaces
- Definition and properties of Lp spaces.
- Convergence of sequence, weak and strong.
- Fourier Analysis.
- Fourier series, Fourier transform;
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.
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:
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.
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.
During the Covid-19 emergency the learning assessment method will consist in a written exam with video surveillance on Webex.
The course includes exercises classes; extra exercises are suggested as homework.
Suggested readings and bibliography
- STROOCK, D. W. Essentials of Integration Theory for Analysis, Springer, 2011
- 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.
- Additional Lecture Notes will be made available to the students.
- Enrollment opening date
- 01/09/2019 at 00:00
- Enrollment closing date
- 30/06/2020 at 00:00