Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space.
Analogous to the finite-dimensional situation, one can define a differentiable Hilbert manifold by considering a maximal atlas in which the transition maps are differentiable.
Many basic constructions of manifold theory, such as the tangent space of a manifold and a tubular neighbourhood of a submanifold (of finite codimension) carry over from the finite dimensional situation to the Hilbert setting with little change.
The reason for this is that Sard's lemma holds for Fredholm maps, but not in general.
Notwithstanding this difference, Hilbert manifolds have several very nice properties.