## Bulletin of the Belgian Mathematical Society - Simon Stevin

### 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.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 4 (2013), 577-586.

Dates
First available in Project Euclid: 22 October 2013

https://projecteuclid.org/euclid.bbms/1382448181

Digital Object Identifier
doi:10.36045/bbms/1382448181

Mathematical Reviews number (MathSciNet)
MR3129060

Zentralblatt MATH identifier
1292.26013

#### Citation

Jiménez-Rodríguez, P.; Muñoz-Fernández, G. A.; Seoane-Sepúlveda, J. B. On Weierstrass' Monsters and lineability. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 4, 577--586. doi:10.36045/bbms/1382448181. https://projecteuclid.org/euclid.bbms/1382448181