  We study a pathwise integral with respect to paths of finite quadratic
variation, defined as the limit of non-anticipative Riemann sums for
gradient-type integrands. We show that the integral satisfies a pathwise
isometry property, analogous to the well-known Ito isometry for stochastic
integrals. This property is then used to represent the integral as a continuous
map on an appropriately defined vector space of integrands. Finally, we obtain
a pathwise 'signal plus noise' decomposition for regular functionals of an
irregular path with non-vanishing quadratic variation, as a unique sum of a
pathwise integral and a component with zero quadratic variation.
