Infinite Dimensional Banach Space of Besicovitch Functions

Abstract

Let $C([0,1])$ be the set of all continuous functions mapping the unit interval $[0,1]$ into $\mathbb{R}$. A function $f\in C([0,1])$ is called Besicovitch if it has nowhere one-sided derivative (finite or infinite). We construct a set $\mathcal{B}_{\sup}\negthickspace\subset C([0,1])$ such that $(\mathcal{B}_{\sup},\vert\vert~\vert\vert_{\sup})$ is an infinite dimensional Banach (sub)space in C([0,1]) and each nonzero element of $\mathcal{B}_{\sup}$ is a Besicovitch function.