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Transactions of the American Mathematical Society

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On the dimension of subspaces of continuous functions attaining their maximum finitely many times


Authors: L. Bernal-González, H. J. Cabana-Méndez, G. A. Muñoz-Fernández and J. B. Seoane-Sepúlveda
Journal: Trans. Amer. Math. Soc. 373 (2020), 3063-3083
MSC (2010): Primary 15A03, 26A15, 46E15
DOI: https://doi.org/10.1090/tran/8054
Published electronically: February 11, 2020
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Abstract: If $ V$ stands for a subspace of $ \mathcal {C}(\mathbb{R})$ such that every nonzero function in $ V$ attains its maximum at one (and only one) point, then we prove that $ \mathrm {dim}(V) \le 2$. This provides the final answer to a lineability problem posed by Vladimir I. Gurariy in 2003. Moreover, we generalize the previous result in the following terms: If $ m \in \mathbb{N}$ and $ V_m$ stands for a subspace of $ \mathcal {C}(\mathbb{R})$ such that every nonzero function in $ V_m$ attains its maximum at $ m$ (and only $ m$) points, then $ \mathrm {dim}(V_m)\le 2$ for $ m > 1$ as well. Besides being a problem closely related to real analysis, this problem actually needs the use of tools from general topology, geometry, and complex analysis, such as decompositions (or partitions) of manifolds or Moore's theorem, among others.


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L. Bernal-González
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Instituto de Matemáticas Antonio de Castro Brzezicki, Universidad de Sevilla, Avenida Reina Mercedes, Sevilla, 41080, Spain
Email: lbernal@us.es

H. J. Cabana-Méndez
Affiliation: Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Ciencias Matemáticas, Plaza de Ciencias 3, Universidad Complutense de Madrid, Madrid, 28040, Spain
Email: hercaban@ucm.es

G. A. Muñoz-Fernández
Affiliation: Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Ciencias Matemáticas, Plaza de Ciencias 3, Universidad Complutense de Madrid, Madrid, 28040, Spain
Email: gustavo_fernandez@mat.ucm.es

J. B. Seoane-Sepúlveda
Affiliation: Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Ciencias Matemáticas, Plaza de Ciencias 3, Universidad Complutense de Madrid, Madrid, 28040, Spain
Email: jseoane@mat.ucm.es

DOI: https://doi.org/10.1090/tran/8054
Keywords: Lineability, spaceability, subspaces of continuous functions, functions attaining their maximum.
Received by editor(s): March 7, 2019
Published electronically: February 11, 2020
Additional Notes: The first author was supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 Grant P08-FQM-03543 and by MCINN Grant PGC2018-098474-B-C21
The second, third, and fourth authors were supported by the Grant MTM2015-65825-P and PGC2018-097286-B-I00
Article copyright: © Copyright 2020 American Mathematical Society