Mathematics

FABIANA LEONI
FABIANA LEONI

General targets: to acquire basic knowledge in measure and integration theory, spaces L^p, Fourier series.

Specific targets: to acquire the ability to use the definitions and theorems contained in the course.

Knowledge and understanding: the student will have acquired the basic notions and results related to the Theory of Abstract Measure, to the construction of the Lebesgue Measure, to the Theory of Integration, to the spaces L^p, to the spaces of Hilbert, to the Fourier series.

Apply knowledge and understanding: the student will be able to understand the concept of measurement and integral in abstract spaces, to integrate discontinuous functions, to operate with different notions of convergence in L^p, to use the Fourier series in L^2.

Critical and judgmental skills: the student will have the basis to deal with some problems of pure and applied mathematics, related to the Theory of Measurement, the spaces L^p, the spaces of Hilbert, the Fourier series.

Communication skills: the student will be able to expose the course contents in a clear and understandable way, both in the written and oral part.

Learning skills: the knowledge acquired will allow study, individual or taught in a master-level course, relating to more specialized aspects concerning the Theory of Measurement, the spaces L ^ p, the spaces of Hilbert, the Fourier series.

The student will have acquired the basic notions and results related to the Theory of Abstract Measure, to the construction of the Lebesgue Measure, to the Theory of Integration, to the spaces L^p, to the spaces of Hilbert, to the Fourier series.
The student will be able to understand the concept of measurement and integral in abstract spaces, to integrate discontinuous functions, to operate with different notions of convergence in L^p, to use the Fourier series in L^2.

We require basic knowledge of calculus in one and more variables. In particular: limits, sequences and series, gradient of functions, Riemann integral, sequences and series of functions, point-wise convergence, local and uniform convergence. Linear algebra and topology.
Not required the students to have passed specific exams in advance.

Measure theory: Lebesgue measure in Rd; General measure and integration theory; Riemann and Lebesgue integral; change of variable formula; product measures, Fubini theorem; L^p spaces.

Hilbert spaces and Fourier series:
Riesz representation theorem; Fourier series.

W. Rudin: “Real & Complex Analysis”
H.L. Royden, P.M. Fitzpatrick : “Real Analysis”

Teaching and didactic materials "Analisi Reale"- L. Fanelli, E. Spadaro

Frontal lectures with the use of blackboard

in room, 7 hours per week

One written exam and one oral exam. Instead of the final written exam, the student can choose to take two
partial exams which will take place at the middle of the course and immediately
after the end, respectively. The final score is given by the average of the scores achieved in the partial exams.

-say if a given set in R^d is Lebesgue measurable and compute its measure

-definition of Lebesgue integral and convergence results for integrals

-put in relationship convergence a.e, a.u., in measure and in L^p for sequences of measurable functions

-point-wise and L^2-convergence of Fourier series