$(\alpha ,\mathfrak{c})$-spaceability of continuous functions with prescribed Hausdorff graph dimension

Abstract

Let $1<s\le 2$ and let $H_s[0,1]$ consist of the real-valued continuous functions on $[0,1]$ whose graphs have Hausdorff dimension $s$. We prove that every finite-dimensional subspace $E\subset C[0,1]$ with $E\setminus \{0\}\subset H_s[0,1]$ is contained in a closed complemented subspace isomorphic to $c_0$, all of whose nonzero elements still belong to $H_s[0,1]$. This settles a question of Liu, Shi and Zhang. We also obtain the exact criterion: $H_s[0,1]$ is $(\alpha ,\mathfrak{c})$-spaceable precisely when $\alpha$ is finite. We also show that every admissible finite-dimensional subspace is contained in a dense subspace of Hamel dimension $\mathfrak{c}$ whose nonzero elements have the same prescribed graph dimension. This also recovers maximal pointwise dense-lineability of $H_s[0,1]$.