On very non-linear subsets of continuous functions
type: Item
instance of: scholarly article
zbMATH Open document ID: 1308.46032
author: D. Cariello
author: G. Botelho
author: V. V. Fávaro
author: Daniel Pellegrino
author: J. B. Seoane-Sepúlveda
published in: The Quarterly Journal of Mathematics
publication date: 2014-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.4395
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 46G25
zbMATH DE Number: 6353664
zbMATH Keywords: lineabiilty
zbMATH Keywords: continuous functions attaining their maximum at only one point
MaRDI profile type: Publication
OpenAlex ID: W2126338783
arXiv ID: 1212.4395
DOI: 10.1093/QMATH/HAT043
title: On very non-linear subsets of continuous functions
review text: A subset \(A \subset E\) of a topological vector space is said to be \(\eta\)-lineable for some cardinal number \(\eta\) if \(A \cup \{0\}\) contains an \(\eta\)-dimensional subspace. In recent years, there has been a lot of work along the following lines: Let \(A\) be the set of functions in \(\mathcal C[0,1]\) having a (usually, strange and anti-intuitive) property. Then, in surprisingly many cases, it turns out that \(A \cup \{0\}\) contains an infinite-dimensional subspace. (See, e.g., the excellent recent survey by \textit{L. Bernal-González} and the fourth and fifth author of this paper in [Bull. Am. Math. Soc., New Ser. 51, No. 1, 71--130 (2014; Zbl 1292.46004)].) It is somehow refreshing to encounter this article, motivated by work of \textit{V. I. Gurariy} and \textit{L. Quarta} [J. Math. Anal. Appl. 294, No. 1, 62--72 (2004; Zbl 1053.46014)], in which properties are studied for which the ``degree of lineability'' \(\eta\) can be shown to be not bigger than 1 or 2.NEWLINENEWLINEThis work is based on three results of Gurariy and Quarta [loc. cit.]: Let \(\hat{\mathcal C}(D)\) denote those \(\mathbb R\)-valued continuous functions \(f\) on the topological space \(D\) such that \(f\) attains its maximum at a unique point of \(D\). Then the following hold:NEWLINENEWLINE (A) \(\hat{\mathcal C}[a,b) \cup \{0\}\) contains a two-dimensional vector space,NEWLINENEWLINE (B) \(\hat{ \mathcal C}(\mathbb R) \cup \{0\}\) contains a two-dimensional vector space, andNEWLINENEWLINE(C) \(\hat{ \mathcal C}[a,b] \cup \{0\}\) does not contain a two-dimensional subspace.NEWLINENEWLINESpecifically, the authors extend (A) to the following consequences of one of their results:NEWLINENEWLINE(A\(^\prime\)) Let \(n \geq 2\) be an integer and \(D\) a topological space admitting a continuous bijection \(D \to S^{n-1}.\) Then \({\mathcal C}(D) \cup \{0\}\) is \(n\)-lineable.NEWLINENEWLINEIn connection with (C), using the Borsuk-Ulam theorem, the authors show:NEWLINENEWLINE (C\(^\prime\))…
reviewed by: Richard M. Aron
description: scientific article; zbMATH DE number 6353664