A weak pseudo-Riemannian metric on a smooth manifold 
modeled on a real Banach space is a smooth 
tensor field 
of continuous symmetric
bilinear forms for which, at each 
,
 |
for all 
implies that 
. Equivalently, 
is an injective linear transformation into the continuous dual normed space 
(Fischer et al. 1980, p. 182). This is weak
non-degeneracy. It does not require
surjectivity, but it does not exclude it. In particular,
every strong pseudo-Riemannian metric
is weak in this sense.
Weak pseudo-Riemannian metrics which are also positive definite are called weak Riemannian metrics. This agrees with the convention that ordinary semi-Riemannian metrics include the positive definite case. In most literature, weak Riemannian metrics are simply called Riemannian.
