  In this paper we consider an abstract Wiener space $(X,\gamma,H)$ and an open
subset $O\subseteq X$ which satisfies suitable assumptions. For every
$p\in(1,+\infty)$ we define the Sobolev space $W_{0}^{1,p}(O,\gamma)$ as the
closure of Lipschitz continuous functions which support with positive distance
from $\partial O$ with respect to the natural Sobolev norm, and we show that
under the assumptions on $O$ the space $W_{0}^{1,p}(O,\gamma)$ can be
characterized as the space of functions in $W^{1,p}(O,\gamma)$ which have null
trace at the boundary $\partial O$, or, equivalently, as the space of functions
defined on $O$ whose trivial extension belongs to $W^{1,p}(X,\gamma)$.
