Abstract
In this paper, we show that the set of continuous functions defined on \({\mathbb {R}}^n\) that approach zero at infinity and attain their maximum at precisely one (and only one) point is n-lineable but not \((n+2)\)-lineable. This result complements some recent published works on an open question originally posed by Vladimir I. Gurariy (1935–2005) in 2003.
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Let E be a topological vector space and \(\alpha \) be a cardinal number. We say that a subset A of E is \(\alpha \)-dense-lineable whenever \(A \cup \{0\}\) contains a dense vector subspace F of E with \(\dim (F) = \alpha .\)
Let E be a topological vector space, and let \(\alpha \) and \(\beta \) be cardinal numbers, with \(\alpha <\beta .\) We say that \(A\subset E\) is \((\alpha ,\beta )\)-lineable if it is \(\alpha \)-lineable and for every subspace \(F_\alpha \subset E\) with \(F_\alpha \subset A\cup \{0\}\) and \(\dim (F_\alpha )=\alpha ,\) there is a closed subspace \(F_\beta \subset E\) with \(\dim (F_\beta )=\beta \) and \(F_\alpha \subset F_\beta \subset A\cup \{0\}.\)
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G. Araújo was supported by Grant 3024/2021, Paraíba State Research Foundation (FAPESQ). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.
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Araújo, G., Barbosa, A. On the set of functions that vanish at infinity and have a unique maximum. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 76 (2024). https://doi.org/10.1007/s13398-024-01573-4
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DOI: https://doi.org/10.1007/s13398-024-01573-4