Definition of normed and inner product space

Question

I was reading some Wikipedia pages about Normed Vector spaces and Inner product spaces and, in the definitions, they always talk about vector spaces over either $\Bbb R$ or $\Bbb C$.

Is this because most of the useful normed and inner product spaces are over $\Bbb R$ or $\Bbb C$ or is those spaces only defined for vector spaces over those specific fields?

Edit: After debating this topic in the comments of this post I want to rephrase my question:

Let $V$ be a vector space over a field $\mathbb F$. What condition should $\Bbb F$ verify if we want $V$ to be able to be an inner product space? How about a normed vector space?

Answer 1

I believe it works over any normed field (at least the normed space, for inner product spaces I'm not sure, since you'd need some generalisation for complex conjugation). A normed field $k$ is a field equipped with a norm $||\cdot||: k\to \mathbb{R}_{\ge0}$ such that

• $||x||=0\Leftrightarrow x=0$
• $||a+b|| \le ||a|| + ||b||$
• $||a\cdot b|| = ||a||\cdot||b||$

If your field $k$ has a discrete valuation $\nu$ that you can build a norm by defining $||x||:=\exp(-a\nu(x))$ for any positive $a$...

In any case, I am sure that Bourbaki will provide you with the most general definition.

And if you want to relax the condition that the norm map to $\mathbb{R}_{\ge0}$, I think there is also a way to do that, and just have it map to some kind of totally ordered semiring...