Uma introdução aos espaços de Lebesgue: completude, separabilidade e reflexibilidade

Abstract

Through this work, I aim to study the properties of separability, reflexivity, completeness, and duality of the spaces Lp (X, Σ, μ) with 1 ≤ p ≤ ∞. For this study, in Chapter 1, we will address preliminary concepts that will serve as a basis in demonstrating future results, highlighting the concept of completeness of a metric space and some of its characteristics. Following this, in Chapter 2, we will discuss what a separable and reflexive space entails, as well as present some of their main properties. Lastly, and not least importantly, for the construction of this work, we will present the study carried out on Lebesgue spaces, aiming to verify the properties of completeness, separability, and reflexivity.