When the norm of a vector is always greater than or equal to zero, the Cauchy-Schwarz inequality holds, but what if we look at a metric with an arbitrary signature? Then the inner product of a vector with itself could be negative. Is there any Cauchy-Schwarz inequality for an arbitrary metric? I suspect it would look something like this:
but I am not sure if that is necessarily the case (let alone how to go about proving it). Here, $g$ is the matrix representation of the metric tensor.