Uma introdução aos espaços de Lebesgue

Abstract

This undergraduate thesis aims to introduce Lebesgue Spaces Lᵖ and demonstrate that they are Banach spaces, that is, we will show that every Cauchy sequence in Lᵖ converges to an element in Lᵖ Initially, basic concepts and results related to the theory of metric spaces will be presented, including the notions of infimum and supremum in the set of real numbers, as well as the definition of the extended real line. Next, an introductory discussion of measure theory will be conducted, covering topics such as measurable functions, measures, and the integral of Lebesgue, establishing the essential theoretical foundations for the formal definition of these spaces. Finally, the main properties of Lᵖ spaces will be explored, such as Hölder's inequality, Minkowski's inequality, and the Riesz-Fischer Theorem.