Some results and open questions on spaceability in function spaces

Abstract

A subset M M of a topological vector space X X is called lineable (respectively, spaceable) in X X if there exists an infinite dimensional linear space (respectively, an infinite dimensional closed linear space) Y ⊂ M ∪ { 0 } Y \subset M\cup \{0\} . In this article we prove that, for every infinite dimensional closed subspace X X of C [ 0 , 1 ] \mathcal {C}[0,1] , the set of functions in X X having infinitely many zeros in [ 0 , 1 ] [0,1] is spaceable in X X . We discuss problems related to these concepts for certain subsets of some important classes of Banach spaces (such as C [ 0 , 1 ] \mathcal {C}[0,1] or Müntz spaces). We also propose several open questions in the field and study the properties of a new concept that we call the oscillating spectrum of subspaces of C [ 0 , 1 ] \mathcal {C}[0,1] , as well as oscillating and annulling properties of subspaces of C [ 0 , 1 ] \mathcal {C}[0,1] .