  We develop a functional Stein-Malliavin method in a non-diffusive Poissonian
setting, thus obtaining a) quantitative central limit theorems for
approximation of arbitrary non-degenerate Gaussian random elements taking
values in a separable Hilbert space and b) fourth moment bounds for
approximating sequences with finite chaos expansion. Our results rely on an
infinite-dimensional version of Stein's method of exchangeable pairs combined
with the so-called Gamma calculus. Two applications are included: Brownian
approximation of Poisson processes in Besov-Liouville spaces and a functional
limit theorem for an edge-counting statistic of a random geometric graph.
