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october 2013 On Weierstrass' Monsters and lineability
P. Jiménez-Rodríguez, G. A. Muñoz-Fernández, J. B. Seoane-Sepúlveda
Bull. Belg. Math. Soc. Simon Stevin 20(4): 577-586 (october 2013). DOI: 10.36045/bbms/1382448181

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.

Citation

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P. Jiménez-Rodríguez. G. A. Muñoz-Fernández. J. B. Seoane-Sepúlveda. "On Weierstrass' Monsters and lineability." Bull. Belg. Math. Soc. Simon Stevin 20 (4) 577 - 586, october 2013. https://doi.org/10.36045/bbms/1382448181

Information

Published: october 2013
First available in Project Euclid: 22 October 2013

zbMATH: 1292.26013
MathSciNet: MR3129060
Digital Object Identifier: 10.36045/bbms/1382448181

Subjects:
Primary: 15A03, 26B05

Rights: Copyright © 2013 The Belgian Mathematical Society

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Vol.20 • No. 4 • october 2013
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