DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCES
Master's Degree in Mathematics
Academic Year 2016/2017 - 1° Year

ELEMENTS OF ADVANCED MATHEMATICAL ANALYSIS

Teaching Staff

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 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 Measure and Integration Theory and consider basic facts in Functional Analysis.

Detailed Course Content

• Module 1
  

  The extended real line: ordering and topology. Sequences of sets. 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.

• Module 2
  

  L^p spaces. Modes 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. Normed and seminormed spaces. Linear Transformations. The Hahn-Banach theorem. Weak and weak* topology.

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