On subspaces of \(C[0, 1]\) consisting of nonsmooth functions (Q2473723)

The author's main result is that for any Hölder space \(H^\omega\), there exists a subspace \(G \subset H^\omega\) such that \(G\) is isomorphic to \(\ell_1\) and such that the restriction of any nonzero function in \(G\) to any set \(D\) of positive measure is not in \(H^\omega(D)\). The technique of proof is constructive. It should be pointed out that there are several relevant complementary references to those cited, containing further results. Among them are \textit{S.\,Hencl} [Proc.\ Am.\ Math.\ Soc.\ 128, No.\,12, 3505--3511 (2000; Zbl 0956.26008)] and \textit{F.\,Bayart} and \textit{L.\,Quarta} [Isr.\ J.\ Math.\ 158, 285--296 (2007; Zbl 1138.46017), reviewed below]. Thus, in fact, there are algebras of badly behaved functions in the sense studied here.