  We prove a local limit theorem, i.e. a central limit theorem for densities,
for a sequence of independent and identically distributed random variables
taking values on an abstract Wiener space; the common law of those random
variables is assumed to be absolutely continuous with respect to the reference
Gaussian measure. We begin by showing that the key roles of scaling operator
and convolution product in this infinite dimensional Gaussian framework are
played by the Ornstein-Uhlenbeck semigroup and Wick product, respectively. We
proceed by establishing a necessary condition on the density of the random
variables for the local limit theorem to hold true. We then reverse the
implication and prove under an additional assumption the desired L1-convergence
of the density of \frac{X_1+...+X_n}{\sqrt{n}}. We close the paper comparing
our result with certain Berry-Esseen bounds for multidimensional central limit
theorems.
