We consider the Kolmogorov equation in infinite dimensions associated with a nonlinear stochastic evolution equation in some Hilbert space. By introducing a proper class of Gauss-Sobolev spaces, aL2-theory of the Kolmogorov equation is developed. Under some suitable conditions, the existence and uniqueness of solutions of the Cauchy problem and the related elliptic problem are proved in such a Gauss-Sobolev space setting