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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Abstract

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

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

Dates
First available in Project Euclid: 6 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1480993591

Mathematical Reviews number (MathSciNet)
MR3579672

Zentralblatt MATH identifier
1357.28004

Subjects
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]

Keywords
lineability open mapping injective function special functions

Citation

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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