Math people:


The title is the question: What fields can be used for an inner product space?

This question has been discussed in Math Stack Exchange with no definitive resolution.  A similar question appeared here, and an answer was accepted, but someone pointed out a serious problem with the answer.  

I am using the standard definition of inner product, which includes $\langle \mathbf{x}, \mathbf{x} \rangle > 0$ for all non zero vectors $\mathbf{x}$.  

It seems to me that any field of prime characteristic does not make sense, because it does not have an order relation that respects addition.  It also seems to me that the field $\mathbb{F}$ can be any ordered field or any subfield of $\mathbb{C}$ that is stable under complex conjugation (for any non-algebraists like me, the word "stable" seems to be standard here. Anyone else would use the word "closed").  I do not know if any other fields are possible.  Of course, an ordered field may or may not be a subfield of $\mathbb{R}$.

It seems to be rare for people, even mathematicians, to use any field other than $\mathbb{R}$ or $\mathbb{C}$ for an inner product space.

Can anyone clear this up?


Stefan (STack Exchange FAN)