Sub-Riemannian geometry on infinite dimensional manifolds

We start from the definition of an infinite-dimensional manifold with a specific choice of the underlying vector space for developing the smooth calculus. Then we define Riemannian and sub-Riemannian structures, and discuss the choice of a tool for studying geodesics on infinite-dimensional sub-Riemannian manifolds. We show that, similarly to the finite-dimensional case, there are two different, but not mutually disjoint classes of geodesics. We present geodesic equations for those classes of geodesics which are natural generalisations of classical Riemannian geodesics.