In this paper we first shall establish an existence and uniqueness result for the semilinear stochastic differential equations in Hilbert space dX=(AX+f(X))dt+g(X)dW under weaker conditions than the Lipschitz one by investigating the convergence of the successive approximations. Secondly we show, under the assumption of so-called pathwise uniqueness (PU), the convergence of the Euler and Lie-Trotter schemes in L^p, p>2 and the continuous dependence of the solutions on the initial data and on the coefficients for such equation. Finally we study the existence of the solutions when the coefficients f and g are only defined on a subset of the state Hilbert space.