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Risorse bibliografiche
Risorsa bibliografica obbligatoria
Risorsa bibliografica facoltativa
Scheda Riassuntiva
Anno Accademico 2018/2019
Scuola Scuola di Ingegneria Industriale e dell'Informazione
Insegnamento 095958 - REAL AND FUNCTIONAL ANALYSIS
Docente Grasselli Maurizio
Cfu 8.00 Tipo insegnamento Monodisciplinare

Corso di Studi Codice Piano di Studio preventivamente approvato Da (compreso) A (escluso) Insegnamento
Ing Ind - Inf (Mag.)(ord. 270) - MI (487) MATHEMATICAL ENGINEERING - INGEGNERIA MATEMATICA*AZZZZ095958 - REAL AND FUNCTIONAL ANALYSIS

Obiettivi dell'insegnamento

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.  


Risultati di apprendimento attesi

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

Argomenti trattati
0. Basic notions of set theory. Equivalence and order relations. Cardinal and ordinal numbers. Axiom of choice. Topological spaces. Metric 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.

 

 


Prerequisiti

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.


Modalità di valutazione

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

Bibliografia
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/

Forme didattiche
Tipo Forma Didattica Ore di attività svolte in aula
(hh:mm)
Ore di studio autonome
(hh:mm)
Lezione
48:00
72:00
Esercitazione
32:00
48:00
Laboratorio Informatico
0:00
0:00
Laboratorio Sperimentale
0:00
0:00
Laboratorio Di Progetto
0:00
0:00
Totale 80:00 120:00

Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua Inglese
Disponibilità di libri di testo/bibliografia in lingua inglese
Possibilità di sostenere l'esame in lingua inglese
schedaincarico v. 1.6.1 / 1.6.1
Area Servizi ICT
28/02/2020