A strong Riemannian metric 
on a smooth manifold 
modeled on a real Banach space, possibly infinite
dimensional, is a smooth assignment of continuous inner products 
to its tangent spaces.
The induced linear transformation
 |
from 
to its dual normed space 
is an isomorphism of
Banach spaces for every 
. This isomorphism and
its inverse are continuous.
Every strong Riemannian metric is a weak Riemannian metric. The two notions coincide in finite dimensions, since an injective linear transformation between vector spaces of the same finite dimension is an isomorphism. In infinite dimensions, a weak Riemannian metric need not be strong (Stacey 2008).
