On Weierstrass' Monsters and lineability

Abstract

Let $E$ be a topological vector space and let us consider a property $\mathcal P$. We say that the subset $M$ of $E$ formed by the vectors in $E$ which satisfy $\mathcal P$ is $\mu$-lineable (for certain cardinal $\mu$, finite or infinite) if $M \cup \{0\}$ contains an infinite dimensional linear space of dimension $\mu$. In 1966 V. Gurariy provided a non-constructive proof of the $\aleph_0$-lineability of the set of {\em Weierstrass' Monsters} (continuous nowhere differentiable functions on $\mathbb{R}$). Here we provide the first constructive proof of the ${\mathfrak c}$-lineability of this set (where $\mathfrak{c}$ denotes the continuum). Of course, this result is the best possible in terms of dimension.