Pathwise space approximations of semi-linear parabolic SPDEs with multiplicative noise

We provide convergence rates for space approximations of semi-linear stochastic differential equations with multiplicative noise in a Hilbert space. The space approximations we consider are spectral Galerkin and finite elements, and the type of convergence we consider is almost sure uniform convergence, i.e. pathwise convergence. The proofs are based on a recent perturbation result for such equations.