We obtain some sufficient conditions for weak convergence of a sequence of processes \(\{X_ n\}\) to \(X\), when \(X\) arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processes \(X_ n\), \(X\) is infinite-dimensional. The usefulness of these conditions is illustrated by deriving Donsker’s invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.