Stochastic Galerkin approximation of operator equations with infinite dimensional noise

It is common practice in the study of stochastic Galerkin methods for boundary value problems depending on random fields to truncate a series representation of this field prior to the Galerkin discretization. We show that this is unnecessary; the projection onto a finite dimensional subspace automatically replaces the infinite series expansion by a suitable partial sum. We construct tensor product polynomial bases on infinite dimensional parameter domains, and use these to recast a random boundary value problem as a countably infinite system of deterministic equations. The stochastic Galerkin method can be interpreted as a standard finite element discretization of a finite section of this infinite system