Introduction to Ordered Vector Spaces

• Grant number: 26/14973-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: September 01, 2026
• End date: August 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Vinícius Colferai Corrêa Miranda
• Grantee: Luiz Gustavo Borges e Cunha
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 25/08630-0 - Infinite dimensional analysis: Banach lattices and geometric measure theory, AP.R

Abstract

This Scientific Initiation project aims to introduce the student to the study of ordered vector spaces, focusing on the fundamental concepts that form the basis for the theory of Riesz spaces, Banach lattices, and positive operators. The proposal consists of a guided study of the first chapter of the book Positive Operators by C. D. Aliprantis and O. Burkinshaw, covering notions such as partial orders compatible with linear structure, positive cones, positive elements, ordered vector spaces, Riesz spaces, and positive operators. The first chapter of the book Banach Lattices by P. Meyer-Nieberg may also be used as complementary material.The theory of ordered vector spaces combines algebraic and order structures, constituting an important tool in modern Functional Analysis. Although this theory has natural developments in more advanced contexts, such as Banach lattices and the theory of positive operators, its initial concepts can be studied abstractly using the language of Linear Algebra. In this sense, the project is suitable for an undergraduate student in their early stages, allowing contact with a modern research area using already familiar mathematical tools.The student is currently in the third semester of the Electrical Engineering program. From a mathematical standpoint, they have already completed the courses Calculus I, Calculus II, Analytic Geometry, and Linear Algebra, showing a particular interest in the study of vector spaces. In addition to providing an introduction to an active research area, the project aims to develop the student's mathematical maturity, familiarity with advanced English texts, and ability to present mathematical results with rigor (AU)