2020/2021

2022/2023

DM270

MAT/05 (MATHEMATICAL ANALYSIS)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

3°

1st semester (29/09/2022 - 13/01/2023)

9

78 lesson hours

Italian

WRITTEN AND ORAL TEST

COLLI PIERLUIGI (titolare) - 9 ECTS

The basics of Mathematical Analysis 1 and 2 and of Linear Algebra are supposed to be known.

The course is divided in two parts and it aims to provide a systematic exposition of the abstract measure theory, with additions on the fundamental theorem of integral calculus, and to present the definitions and first results on normed spaces, Banach and Hilbert spaces, also discussing projections and abstract Fourier series. The theory is accompanied by examples and exercises.

Measure theory. Lebesgue measure, measurable sets and functions, Lebesgue integral, passage to the limit under the integral, different types of convergence.

Product measures, Fubini and Tonelli theorems. Real measures, Hahn decomposition, Radon-Nikodym theorem, functions with bounded variation.

Normed spaces and Banach spaces. Linear continuous operators. L^p spaces with their properties.

Hilbert spaces, Riesz and projections theorems, Fourier series.

Extended summary

Measure theory. Lebesgue measure, sigma-algebras, measures, measurable functions, Lebesgue integral, theorems of passage to the limit under the integral, almost-everywhere and quasi-uniform convergences, convergence in measure.

Product measures, Fubini and Tonelli theorems. Real measures, Hahn decomposition, absolutely continuous measures, Radon-Nikodym theorem, functions of bounded variation, absolutely continuous functions and the fundamental theorem of calculus.

Normed spaces and Banach spaces: foundations of the theory. Subspaces. Linear continuous operators. Dual space. Numerous examples. L^p spaces with their properties: the Young, Hölder, Minkowski inequalities. Completeness.

Hilbert spaces: Riesz and projections theorems. Fourier series: decomposition theorems, complete orthonormal systems, Riesz-Fisher theorem. Fourier series in L ^ 2_T and completeness of the system exp (ikT). Convolutions with trigonometric polynomials and Fejer kernels.

Lectures and exercises in the classroom, largely run on the blackboard. Availability to discuss with students during reception hours.

G. Gilardi: Analisi Matematica di Base, McGraw-Hill
G. Gilardi: Analisi 3, McGraw-Hill
H. Brezis: Functional Analysis, Springer
in addition to the educational resources available on the course web page.

The exam consists of a written test with 2 hours of time (during which it is not allowed the use of notes, texts, minicomputers, ...) plus oral examination. The result of the written test is not binding to participate in the oral test and the success of the examination, but of course it is an important element of judgment for the final evaluation.

The teachers are available to the students to provide them with indications and suggestions for the selection of texts and educational material, as well as proposals for exercises, exam tests and theoretical support material.