Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover $$ \mathcal{U} $$ of X there is a sequence of maps (f n : X → X) nεgw such that each f n is $$ \mathcal{U} $$ -near to the identity map of X and the family {f n (X)} n∈ω is locally finite in X. Also we show that a metrizable space X of density dens(X) < $$ \mathfrak{d} $$ is a Hilbert manifold if X has gw-LFAP and each closed subset A ⊂ X of density dens(A) < dens(X) is a Z ∞-set in X.