Infinite dimensional Banach space of Besicovitch functions. (English) Zbl 1213.46024

Bobok, Jozef

Real Anal. Exch. 32(2006-2007), No. 2, 319-333 (2007).

Summary: 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 \(B_{\sup}\negthickspace\subset C([0,1])\) such that \((B_{\sup},\|\,~\,\|_{\sup})\) is an infinite dimensional Banach (sub)space in \(C([0,1]\)) and each nonzero element of \(B_{\sup}\) is a Besicovitch function.

MSC:

46E15	Banach spaces of continuous, differentiable or analytic functions
26A27	Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives

Keywords:

Besicovitch function; spaceability