Bulletin of the Belgian Mathematical Society - Simon Stevin

On Weierstrass' Monsters and lineability

P. Jiménez-Rodríguez, G. A. Muñoz-Fernández, and J. B. Seoane-Sepúlveda

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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

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

First available in Project Euclid: 22 October 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A03: Vector spaces, linear dependence, rank 26B05: Continuity and differentiation questions

lineability spaceability continuous nowhere differentiable function Weierstrass' Monster


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

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