O Teorema do Ponto Fixo de Banach e a sua relação com alguns resultados de Análise Funcional

Abstract

In this work, we will address the Banach Fixed Point Theorem and apply it to prove some classical results in Functional Analysis, such as Stampacchia’s Theorem and, as a consequence, the Lax-Milgram Theorem. Additionally, we will study compact operators between Banach spaces and show how we use the Banach Fixed Point Theorem to obtain some properties of the spectrum of such operators. In this context, we also present the famous results on compact operators: Fredholm’s Alternative and the Spectral Theorem in Hilbert spaces.