The Hopf-Rinow theorem in infinite dimension
Ivar Ekeland · Journal of Differential Geometry · 1978
In this paper, the following theorem is proven. Let M be a complete, possibly infinite-dimensional, Riemannian manifold. Take any point q in M. Then the set of points p epsilon M which can be joined to q by a unique minimal geodesic contains a countable intersection of open dense subsets.