Infinite-dimensional geometry 4 2.: The geodesic distance of a Riemannian manifold

We describe the geodesic distance of a weak Riemannian manifold. If the Riemannian manifold is strong, the geodesic distance yields a well defined distance function on the manifold.
We present several examples of weak Riemannian manifolds for which this is not the case (and even worse the distance vanishes identically). Thus in infinite-dimensions the geodesic distance will in general only be a pseudodistance.
This is part of the course MAT331: Topics in Analysis at the University of Bergen in Fall 2020.