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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Lineability and spaceability of sets of functions on $\mathbb{R} $


Authors: Richard Aron, V. I. Gurariy and J. B. Seoane
Journal: Proc. Amer. Math. Soc. 133 (2005), 795-803
MSC (2000): Primary 26A27, 46E10, 46E15
Published electronically: August 24, 2004
MathSciNet review: 2113929
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Abstract: We show that there is an infinite-dimensional vector space of differentiable functions on $\mathbb{R} ,$ every non-zero element of which is nowhere monotone. We also show that there is a vector space of dimension $2^c$ of functions $\mathbb{R}\to \mathbb{R} ,$ every non-zero element of which is everywhere surjective.


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Additional Information

Richard Aron
Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44242
Email: aron@math.kent.edu

V. I. Gurariy
Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44242
Email: gurariy@math.kent.edu

J. B. Seoane
Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44242
Email: jseoane@math.kent.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07533-1
PII: S 0002-9939(04)07533-1
Received by editor(s): March 26, 2003
Received by editor(s) in revised form: October 28, 2003
Published electronically: August 24, 2004
Additional Notes: The author thanks Departamento de Matemáticas of the Universidad de Cádiz (Spain), especially Antonio Aizpuru, Fernando León, Javier Pérez, and the rest of the members of the group FQM-257.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2004 American Mathematical Society