@Gerald: I don't even care for this question whether square roots of $\langle \mathbf{x},\mathbf{x}\rangle$ exist, or whether you can carry out the Gram-Schmidt process, just the basic definition of inner product (with positive-definiteness).

@Gerald : Thanks. I don't really care about metric completeness for this question. Either I forgot to omit that part from the Wikipedia quote, or someone put it back in. What about the mysterious "additional structure, such as a distinguished automorphism", that the Wikipedia article refers to? Do we need to impose that, is it there already, or is the article wrong?

@Gerald : regarding your suggestion, if we take a formally real field, we can just impose an order on it and make it an ordered field. Why can't we keep the condition $\langle \mathbf{x}, \mathbf{x} \rangle > 0$ for nonzero $\mathbf{x}$? And do we have the "additional structure" that wikipedia says we need (please see my edited question)?

@darij: regarding your first comment, I just reread Wikipedia's article on formally real fields and they mention an alternative definition concerning $-1$ being a sum of squares, so I understand your first comment now.

@darij : Thanks. Apparently "ordered field" refers to a field together with the order relation while a "formally real" field is a field that can be endowed with an order relation with the required properties. I don't really understand your first comment. I don't see an obvious order relation for formal real-valued rational functions in real variables $x$ and $y$ similar to the standard order relation for rational functions in just $x$.

@Mark Grant : Thanks. I guess I didn't read the Wikipedia article closely enough. The article is helpful but not 100% satisfactory. I edited my question.

@Gerald : thanks, I never heard of a "formally real field". I read the link, and it does not explain whether there is any difference between "formally real field" and "ordered field", that is, it is unclear whether there exists a field that belongs to one class and not the other. In addition, I want to keep the requirement $\langle \mathbf{x},\mathbf{x}\rangle > 0$ for all nonzero vectors $\mathbf{x}$.

@Mark Grant : not really. I have read that article several times and I just looked at it again. They seem to use only $\mathbb{R}$ or $\mathbb{C}$ until they discuss "generalizations", of inner product spaces , and I am not interested in generalizations.