Um estudo comparativo entre espaços vetoriais normados de dimensão finita e infinita

Abstract

Vector spaces are structures in which we can add elements and multiply their elements by scalars. When a vector space is provided with a norm, we can also verify metric and topological properties. In Linear Algebra, we study important results that hold for finite-dimensional vector spaces. However, we cannot always extend these results to infinite-dimensional normed vector spaces. With the help of Linear Algebra, Metric Spaces and Functional Analysis, we will see basic notions and enough tools to discuss some differences between normed vector spaces of finite and infinite dimensions. The differences we’ll see are related to norms, linear transformations, completeness, compactness, and closed vector subspaces. We will show valid results for finite dimensional spaces and present examples and counterexamples to show that such results are not always valid in infinite dimensions.