On very non-linear subsets of continuous functions (Q2922452)

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\(^\prime\)) For every compact subset \(D \subset \mathbb R^m\), the set \(\hat{ \mathcal C}(D) \cup \{0\}\) is not \(n\)-lineable for any \(n > m\).NEWLINENEWLINE On the other hand, an example is given of an infinite-dimensional compact set \(D\) such that \(\hat {\mathcal C}(D) \cup\{0\}\) contains an infinite-dimensional subspace.