  We introduce a framework to derive quantitative central limit theorems in the
context of non-linear approximation of Gaussian random variables taking values
in a separable Hilbert space. In particular, our method provides an alternative
to the usual (non-quantitative) finite dimensional distribution convergence and
tightness argument for proving functional convergence of stochastic processes.
We also derive four moments bounds for Hilbert-valued random variables with
possibly infinite chaos expansion, which include, as special cases, all
finite-dimensional four moments results for Gaussian approximation in a
diffusive context proved earlier by various authors. Our main ingredient is a
combination of an infinite-dimensional version of Stein's method as developed
by Shih and the so-called Gamma calculus. As an application, rates of
convergence for the functional Breuer-Major theorem are established.
