On the Riemann integrability of the norm of a path in normed spaces

Abstract

A useful result is that if a bounded complex-valued path is Riemann-integrable, then its modulus is also Riemann-integrable. The extension of this last result to bounded paths taking values in a normed space is affirmed, as being true, in [3]. However, we show that if a bounded path taking values in a normed space is barely Riemann-integrable, then it is not really guaranteed that the norm of this path is also Riemann-integrable.