On Weierstrass' Monsters and lineability

Abstract

Let E be a topological vector space and let us consider a property P. We say that the subset M of E formed by the vectors in E which satisfy 2 is mu-lineable (for certain cardinal mu, finite or infinite) if M U {0} contains an infinite dimensional linear space of dimension mu. In 1966 V. Gurariy provided a non-constructive proof of the N-0-lineability of the set of Weierstrass' Monsters (continuous nowhere differentiable functions on R). Here we provide the first constructive proof of the c-lineability of this set (where c denotes the continuum). Of course, this result is the best possible in terms of dimension.