Hilbert spaces require a positive definite inner product and are very well studied. Have vector spaces with a real "inner product" been studied at all, and if so under what name? In other words $\langle x|y\rangle $ is a real number, not just positive definite, and $\langle x|x\rangle = 0$ does not imply $|x\rangle = 0$. A search of the internet thus far has turned up nothing.
Motivation: The study of hilbert spaces grew out of the concepts of vectors in 3D euclidean space with a positive definite inner product of those vectors. It is well known today that spacetime is non-euclidean and at least 4-dimensional with a real metric, not a positive definite one like euclidean spaces. There are some results for Riemannian manifolds with "pseudo-riemannian", non-positive definite metrics but so far I can't find any studies of vector spaces with such a property for the "inner product" on that space.
Clearly the $\langle x|x\rangle = 0$ for non-zero $|x\rangle$ would be the toughest axiom to deal with. Perhaps it is so tough that no one has tackled it or been able to come up with any useful results. It would seem though that the clear importance of real metrics in the physical universe would make it an extremely important area to investigate.