[internet555]

For one example the notion of a generalized inner product and its induced norm is important for the derivation of conjugate gradient algorithm. My smarter friends have told me that anything involving cost functions and covariance matrices (as might arise in a log likelihood thing for a multivariate normal random variable) will probably end up with weighted norms as well, but I have likely mangled this. So to mangle things further I think that this is very important to many fields eg control or statistics