Ing Ind - Inf (Mag.)(ord. 270) - MI (487) MATHEMATICAL ENGINEERING - INGEGNERIA MATEMATICA

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095958 - 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 Analysishttp://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