Bulletin of the Belgian Mathematical Society - Simon Stevin

Injective mappings in $\mathbb{R}^\mathbb{R}$ and lineability

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

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It is known that there is not a two dimensional linear space in $\mathbb R^\mathbb R$ every non-zero element of which is an injective function. Here, we generalize this result to arbitrarily large dimensions. We also study the convolution of non-differentiable functions which gives, as a result, a differentiable function. In this latter case, we are able to show the existence of linear spaces of the largest possible dimension formed by functions enjoying the previous property. By doing this we provide both positive and negative results to the recent field of lineability. Some open questions are also provided.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 23, Number 4 (2016), 609-623.

First available in Project Euclid: 6 December 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
Secondary: 15A03: Vector spaces, linear dependence, rank 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]

lineability open mapping injective function special functions


Jiménez-Rodríguez, P.; Maghsoudi, S.; Muñoz-Fernández, G.A.; Seoane-Sepúlveda, J.B. Injective mappings in $\mathbb{R}^\mathbb{R}$ and lineability. Bull. Belg. Math. Soc. Simon Stevin 23 (2016), no. 4, 609--623. https://projecteuclid.org/euclid.bbms/1480993591

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