Advanced Topics in Functional Analysis

• Grant number: 25/13277-8
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: November 01, 2025
• End date: October 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Suzete Maria Silva Afonso
• Grantee: Wellington Moisés de Oliveira
• Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil

Abstract

This Scientific Initiation project aims to deepen the study of fundamental and advanced topics in Functional Analysis, with emphasis on weak and weak- topologies, the spectral theory of compact and self-adjoint operators, topological vector spaces, basic techniques in nonlinear analysis, and elements of the theory of Banach spaces. Understanding weak topologies is essential for studying convergence in function spaces and for analyzing optimization problems and differential equations. The spectral theory of operators is fundamental in the formulation of physical and mathematical problems involving eigenvalues and the structure of the associated spaces. The study of topological vector spaces expands the ability to work in more general contexts than those provided by normed spaces, while nonlinear analysis opens avenues for addressing highly relevant problems in applied sciences, where linearity cannot be assumed. Finally, a deeper understanding of the structure of Banach spaces enables the mastery of fundamental tools for various areas of contemporary mathematics. This project will contribute to the student's solid theoretical training, aiming to prepare him for more advanced research activities.