Espaços métricos: uma generalização do conceito de distância - Holanda, Marcus Italo Tavares (2023)
This work consists of a study of the basic aspects of the important theory of metric spaces to shed light on the process of generalizing the concept of distance with the definition of metric, and on relevant related topics such as geometry in metric spaces, the study of continuity of applications and rudiments of topology in these spaces as well as the study of notions of compactness with its characterization in complete metric spaces and the demonstration of the Tychonoff and Ascoli-Arzelá theorems. The work in its culmination presents applications of this study in the answer to two major problems, which are the existence and uniqueness of fixed points in complete metric spaces and the existence and uniqueness of local solutions of ordinary differential equations, with the demonstration of the fixed point theorem of Banach and the Picard-Lindelöf Theorem on the existence and uniqueness of ODEs.