Bibliographic resources
 Bibliography mandatory Bibliography not mandatory
 Summary Teaching Assignment
 Academic Year 2019/2020 School School of Industrial and Information Engineering Course 095958 - REAL AND FUNCTIONAL ANALYSIS Cfu 8.00 Type of Course Mono-Disciplinary Course Lecturers: Titolare (Co-titolari) Grasselli Maurizio

Programme Track From (included) To (excluded) Course
Ing Ind - Inf (Mag.)(ord. 270) - MI (487) MATHEMATICAL ENGINEERING - INGEGNERIA MATEMATICA*AZZZZ095958 - REAL AND FUNCTIONAL ANALYSIS

 Goals
 This course aims to provide students with the knowledge of some basic topics of Mathematical Analysis. Such topics play an essential role, for instance, in Probability and Statistics (measure theory and integration) as well as in the modern approach to partial differential equations (Banach and Hilbert spaces, linear operators). The goal is to get students familiar not only with the results and their applications, but also with the technicalities of the proofs and the rigorous check of the hypotheses  through examples and counterexamples.

 Expected learning outcomes
 Lectures and exercise sessions will allow students to acquire the following competences:   Knowledge and understanding   know some basic concepts of mathematical analysis which are helpful in several applications know the proofs of some fundamental theorems in measure and integration theory and in linear functional analysis understand the range of applicability of various results through counterexamples   Ability in applying knowledge and understanding   state rigorous definitions of the presented notions state and prove the most relevant theorems solve theoretical problems related to the presented theory   Making judgements   find simple counterexamples   Communication skills   write and speak mathematical concepts in a clear and rigorous way

 Topics
 0. Basic notions of set theory. Equivalence and order relations. Cardinal and ordinal numbers. Axiom of choice. Metric and topological spaces. Continuous and semicontinuous functions.   1. Measure space and measurable functions. Positive measures and measurable spaces. Lebesgue measure. Abstract integration. Comparison between Lebesgue and Riemann integrals. Convergences. Derivative of a measure and Radon-Nikodym theorem. Fundamental theorems of Calculus. Product measures and Fubini-Tonelli theorem.   2. Normed spaces and Banach spaces. Spaces of integrable functions. Linear operators. Dual spaces and Hahn-Banach theorem. Weak convergences. Compact operators.   3. Hilbert spaces. Scalar product and its consequences. Riesz representation theorem. Orthonormal basis. L^2 space and Fourier series. Spectral theorem for compact symmetric operators. Fredholm alternative.

 Pre-requisites
 Students are required to know the following topics: real numbers, the euclidean n-dimensional space, functions of real variables and their basic properties, limits of functions of real variables, the main results about differentiation and integration of functions of real variables, sequences and series of numbers and functions, basics of the theory of ordinary differential equations.

 Assessment
 The exam consists of a written test (compulsory) and an oral examination. The written test consists of two parts denoted, respectively, A and B. Part A contains theoretical questions in order to ascertain the understanding of the basic notions (definitions, theorems, proofs, counterexamples) of measure theory, abstract integrations and linear functional analysis. Part B contains some exercises whose solutions require the theoretical tools developed through the course. The maximum grade of A+B is 30/30. In order to pass the exam, the student must get at least 18/30. The oral exam is decided case by case. For instance, if a student gets 16/30 or 17/30 in A+B then she/he is allowed to take the oral exam. In any case, a student who gets a sufficient grade in A+B (i.e. at least 18/30) can ask to take an oral exam. To get the laude the oral exam is mandatory. No mid-term test is planned.   The exam has the goal of checking whether the student has acquired the following skills:   knowledge of some basic concepts of mathematical analysis which are helpful in several applications knowledge of the proofs of some fundamental theorems in measure and integration theory and in linear functional analysis understanding of the range of applicability of various results through counterexamples ability to state rigorous definitions of the presented notions ability to prove some of the presented theorems ability to apply the presented theory to solve given problems  finding simple counterexamples communicating mathematical concepts in a clear and rigorous way

 Bibliography
 A. Bobrowski, Functional Analysis for Probability and Stochastic Processes, Editore: Cambridge University Press, Anno edizione: 2005 E. Hewitt, K.Stromberg, Real and Abstract Analysis, Editore: Springer, Anno edizione: 1975 A.N. Kolmogorov, S.V. Fomin, Introductory Real Analysis, Editore: Dover, Anno edizione: 1975 V. Pata, Appunti del corso di Analisi Reale e Funzionale H.L. Royden, P. Fitzpatrick, Real Analysis, Editore: Pearson, Anno edizione: 2010 G. Teschl, Topics in Real and Functional Analysis http://www.mat.univie.ac.at/~gerald/ftp/book-fa/

 Software used
 No software required

 Learning format(s)
Type of didactic form Ore di attività svolte in aula
(hh:mm)
Ore di studio autonome
(hh:mm)
Lesson
52:00
78:00
Training
28:00
42:00
Computer Laboratory
0:00
0:00
Experimental Laboratory
0:00
0:00
Project Laboratory
0:00
0:00
Total 80:00 120:00

 Information in English to support internationalization
 Course offered in English Textbook/Bibliography available in English It is possible to take the examination in English
 schedaincarico v. 1.10.0 / 1.10.0 Area Servizi ICT 15/07/2024