  In this work we are concerned with the study of the strong order of
convergence in the averaging principle for slow-fast systems of stochastic
evolution equations in Hilbert spaces with additive noise. In particular the
stochastic perturbations are general Wiener processes, i.e their covariance
operators are allowed to be not trace class. We prove that the slow component
converges strongly to the averaged one with order of convergence $1/2$ which is
known to be optimal. Moreover we apply this result to a slow-fast stochastic
reaction diffusion system where the stochastic perturbation is given by a white
noise both in time and space.
