ELEMENTS OF ADVANCED MATHEMATICAL ANALYSIS

Academic Year 2018/2019 - 1° Year - Curriculum APPLICATIVO, Curriculum DIDATTICO and Curriculum TEORICO

Learning Objectives

• Module 1

  The aim of the course is to make the students familiar with basic concepts, main theorems and most used techniques in Measure and Integration Theory and in Functional Analysis. This will give the students a more complete education in the field of Mathematical Analysis and will provide them with useful prerequisites in order to follow more advanced courses. In particular Module 1 focuses on the basic facts of Measure and Integration Theory.

• Module 2

  The aim of the course is to make the students familiar with basic concepts, main theorems and most used techniques in Measure and Integration Theory and in Functional Analysis. This will give the students a more complete education in the field of Mathematical Analysis and will provide them with useful prerequisites in order to follow more advanced courses. In particular Module 2 both deepens some aspects of Real Analysis and consider basic facts in Functional Analysis.

Course Structure

• Module 1

  The course main topics will be explained by the teacher during formal lectures. These will be focusing on each topic's general principles and new concepts that have not been studied before. Each's topic's additional resources and subchapters will be presented by turning over groups of students. The goal is to have students develop study autonomy and teaching abilities, skills that are essential for students who want to pursue a career in research or teaching.

• Module 2

  The course main topics will be explained by the teacher during formal lectures. These will be focusing on each topic's general principles and new concepts that have not been studied before. Each's topic's additional resources and subchapters will be presented by turning over groups of students. The goal is to have students develop study autonomy and teaching abilities, skills that are essential for students who want to pursue a career in research or teaching.

Detailed Course Content

• Module 1

  Lebesgue measure. Measures, outer measure and Carathéodory's theorem. Borel sets of a topological space. Borel measures and distribution functions.Completion of a measure space. Measurable funcions. Sets which are not Lebesgue measurable and Lebesgue measurable sets that are not Borel sets. Signed measures. Integration in a measure space. L^p-spaces. Various types of convergence of sequences of measurable functions. Measure with density and Radon-Nikodym theorem. A characterization of convergence in L^p:Vitali's theorem. Product measure and Fubini's theorem.

• Module 2

  Bounded variation and absoluyely continuous real functions. Normed and seminormed spaces. Linear Transformations. The Hahn-Banach theorem. Weak and weak* topology. Hilbert spaces. Fourier transform.

Textbook Information

• Module 1

  1. A. Villani, Appunti del corso di Istituzioni di Analisi Superiore, lecture notes on line

  2. W. Rudin, Real and Complex Analysis, Third edition, Mc Graw Hill

• Module 2

  1. A. Villani, Appunti del corso di Istituzioni di Analisi Superiore, dispense on line

  2. W. Rudin, Real and Complex Analysis, Third edition, Mc Graw Hill

  3. R. Larsen, Functional Analysis, an introduction, Marcel Dekker

  4. H. Royden, Real Analysis, 2nd ed., Macmillan