A nonlinear Hilbert-space-valued stochastic differential equation where L-1 (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L-1, the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L-1 is not nuclear, a solution in most cases lies not in H but in a larger Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove under suitable conditions that a unique strong solution can still be found to lie in the space H itself. Uniqueness of the weak solution is proved without moment assumptions on the initial random variable,A second problem considered is the asymptotic behavior of the sequence of empirical measures determined by the solutions of an interacting system of H-valued diffusions. It is shown that the sequence converges in probability to the unique solution Lambda(0) of the martingale problem posed by the corresponding McKean-Vlasov equation.