Mathematical analysis, differentiation, Lebesgue integral and functional

• Grant number: 17/26381-1
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: April 01, 2018
• End date: March 31, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Hildebrando Munhoz Rodrigues
• Grantee: André Gustavo Espinhosa Coladello
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

The general objective of this project is the study of Mathematical Analysis, with the purpose of deepening students' knowledge of mathematics and giving them a solid foundation for better development of their academic and professional lives. The seminars will be presented in English, alternately by each student. In addition to introducing numerical sets and their properties, Basics of Metric Spaces Topology, Continuous Functions, Differentiable Functions, Sequences and Numerical and Functions Series. Then theory of measurement and integration, more specifically the Lebesgue integral and its applications, such as integrals of dependent functions parameters and derivation under the integration signal. For this purpose, the books studied will be Walter's Principles of Mathematical Analysis, Walter Rudin, and A Integral de Lebesgue, by C. S. H onig. This will serve as a basis for the second year of a future program that can be started in which topics of operator theory will be studied infinite-dimensional vector spaces, ranging from basic notions of spaces vectors and normed vector spaces, to positive matrices. In this module the book A Short Introduction to Perturbation Theory for Linear Operators, by T. Kato will be mainly used. We will use other reference books as appropriate, who se relationship is presented in the bibliographic references. In addition, the participants should familiarize themselves with the AMS-LATEX software, which will beused for the preparation of the reports.