Generalization of inner product I was wondering if there was a widely accepted generalization of inner product spaces where the inner product look something like $\langle\bullet  , \bullet\rangle:V\times V \to \mathbb{F}$, where $\mathbb{F}$ does not have to equal to $\mathbb{R}$ or $\mathbb{C}$. Could you define a meaningful inner product space over $\mathbb{Q}$ or a finite field?

This generalizes to a sesquilinear form on any module over a ring that admits an antiautomorphism.  See Sesquilinear form on Wikipedia.  The positive-definiteness of an inner product gets replaced by a constraint to a particular subset of the ring.
