Completeness in the Space of Continuous Functions Rn

Abstract

The discipline of metric spaces is present in most of the undergraduate courses in
Mathematics, both undergraduate and baccalaureate. In this discipline, properties related to the
completeness of a space are studied, with usual examples being discussed as R and Rn, but when
studying applications it is necessary to use more spaces functions, such as function space. To study these spaces more deeply, concepts related to Functional Analysis are necessary, since this space has an infinite dimension. However, using more basic concepts seen in Metric Spaces, we can study the space of continuous real functions defined in a range [a, b], space denoted by C[a, b]. When working in this space, we have that each point is a function f : [a, b] → R, that is, functions are treated as metric space points. However, our focus is on how two metrics in the same space can change the characteristics of this space, such as completeness. Thus, we conclude that two different metrics in the space of this space verifying that, when we change the
metric, we can lose interesting properties like completeness.