On (non)lineability and (non)spaceability in L1 spaces

Abstract

In the present dissertation, we will provide methods for constructing lineable or even spaceable sets in certain Banach spaces. Our approach will also allow us to exhibit examples of sets that are lineable but not spaceable, and sets that are not even lineable. Let v = (vn) be an element of ℓ1 with finitely many zero entries. In this setting, we will discuss the lineability (spaceability) of the following sets: B(v) (resp. A0(v)) the set of all elements of ℓ1 where the limit comparison test with respect to v works (resp. fails); X(v) the set of all elements of ℓ1 where the standard comparison test fails (with respect to v). On this matter, we will prove that the set A0(v) is c-dense-lineable but not spaceable. Meanwhile, the set B(v) ∪ {0} only contains finite-dimensional subspaces of dimension 1. Moreover, our methods will allow us to conclude: (1) every infinite-dimensional closed subspace of ℓ1 contains an element of X(v); (2) X(v) is c-dense-lineable and c-spaceable. As an application of our above mentioned findings, we will be able to conclude that the set formed by all elements of ℓ1 for whose generated series the root (resp. ratio) test fails is spaceable. In addition, we will also retrieve some known results. For instance, we will prove that every infinite-dimensional closed subspace of ℓ1 contains an element with infinitely many zeros. At the end, we will extend some of these results to the case L1(M), where M is an unbounded subset of a fixed normed vector space Y, and M is equipped with the Borel σ-algebra.