On Subspaces of $C[0,1]$ Consisting of Nonsmooth Functions

For each Hölder space $H^\omega$, we construct an infinite-dimensional closed subspace $G$ of $C[0,1]$, isomorphic to $l^1$ and such that, for each function $x\in G$ not identically zero, its restriction to the set of positive measure does not belong to the Hölder space $H^\omega$.