ID:
502225
Duration (hours):
84
CFU:
9
SSD:
ANALISI MATEMATICA
Located in:
PAVIA
Year:
2025
Overview
Date/time interval
Primo Semestre (25/09/2025 - 14/01/2026)
Syllabus
Course Objectives
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. The expected learning outcomes are the understanding and knowledge of the topics covered in class for measure theory and function spaces and the ability to construct examples and solve problems related to the subject covered.
Course Prerequisites
The basics of Mathematical Analysis 1 and 2 and of Linear Algebra are supposed to be known.
Teaching Methods
Lectures and exercises in the classroom, largely run on the blackboard. Availability to discuss with students during reception hours.
Assessment Methods
The exam consists of a written part and an oral part. In the written part (during which the use of notes, texts, minicomputers, ... is not permitted), lasting no more than 2 hours,
the skills that the student has achieved in understanding and solving problems regarding the course topics are evaluated. The exercises will be structured so as to cover the different topics developed during the course.
The outcome of the written test is not binding for participation in the oral test and the success of the exam, but obviously constitutes an important element of judgment for the final evaluation.
In the oral part, we will focus above all on verifying the level of knowledge of the notions presented during the course, on the clarity of the presentation and on the student's ability to apply the learned notions.
The formulation of the grade will be obtained by comparing the evaluation of the written part and the oral part.
the skills that the student has achieved in understanding and solving problems regarding the course topics are evaluated. The exercises will be structured so as to cover the different topics developed during the course.
The outcome of the written test is not binding for participation in the oral test and the success of the exam, but obviously constitutes an important element of judgment for the final evaluation.
In the oral part, we will focus above all on verifying the level of knowledge of the notions presented during the course, on the clarity of the presentation and on the student's ability to apply the learned notions.
The formulation of the grade will be obtained by comparing the evaluation of the written part and the oral part.
Texts
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.
G. Gilardi: Analisi 3, McGraw-Hill
H. Brezis: Functional Analysis, Springer
in addition to the educational resources available on the course web page.
Contents
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.
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.
Course Language
Italian
More information
The teacher is 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. Students in the categories identified by the project on innovative teaching will have the opportunity to hold receptions also online and by appointment at times to be agreed with the teacher, or view the teacher's lecture notes.
Degrees
Degrees
MATHEMATICS
Bachelor’s Degree
3 years
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