In a vector space, the scalar product has the property that $\langle v|v\rangle\ge 0$, where $0$ is the additive identity of that field. We use "greater than" and "equal" sign because the field over which vector space is defined satisfy ordering axion (ordered field).
If we define vector space over an unordered field (I don't see any example of it though but I think it can be defined as the axioms of vector space don't require any ordering property), then we don't define the scalar product, because then it won't satisfy the above inequality.
I have two questions-
(i) So, the scalar product is like irrelevant of the vector space (like it is not true that "whenever a vector space is defined, scalar product is also defined")?
(ii) Also the norm of vector also not defined always?