  We prove that the L^2 Riemannian metric on the manifold of all smooth
Riemannian metrics on a fixed closed, finite-dimensional manifold induces a
metric space structure. As the L^2 metric is a weak Riemannian metric, this
fact does not follow from general results. In addition, we prove several
results on the exponential mapping and distance function of a weak Riemannian
metric on a Hilbert/Frechet manifold. The statements are analogous to, but
weaker than, what is known in the case of a Riemannian metric on a
finite-dimensional manifold or a strong Riemannian metric on a Hilbert
manifold.
