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

Programme Track From (included) To (excluded) Course


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

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.




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.


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

Risorsa bibliografica facoltativaA. Bobrowski, Functional Analysis for Probability and Stochastic Processes, Editore: Cambridge University Press, Anno edizione: 2005
Risorsa bibliografica facoltativaE. Hewitt, K.Stromberg, Real and Abstract Analysis, Editore: Springer, Anno edizione: 1975
Risorsa bibliografica facoltativaA.N. Kolmogorov, S.V. Fomin, Introductory Real Analysis, Editore: Dover, Anno edizione: 1975
Risorsa bibliografica obbligatoriaV. Pata, Appunti del corso di Analisi Reale e Funzionale
Risorsa bibliografica facoltativaH.L. Royden, P. Fitzpatrick, Real Analysis, Editore: Pearson, Anno edizione: 2010
Risorsa bibliografica facoltativaG. 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
Ore di studio autonome
Computer Laboratory
Experimental Laboratory
Project Laboratory
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