Just like there are positive real numbers. Does there exist a analogous part to that?
The reason I ask is because, I was studying Inner product spaces where the fields are either $\mathbb{R}$ or $\mathbb{C}$ .
When I ask my teacher why only we define the field to be either $\mathbb{R}$ or $\mathbb{C}$ . He says inner product is generally use to determine the angle between the vectors and their length. And we understand the concept of length and angles in only real numbers.
But why not others example, $\mathbb{Z}_p^2$ is a field over itself ($\mathbb{Z_p}$) where p is prime. Why can't we define an inner product which maps $\mathbb{Z}_p^2\times\mathbb{Z}_p^2$ to $\mathbb{Z}_p$.
Even if we do there comes two properties which inner product has to satisfy by definition_
Positivity_ which I don't know or is there even a thing analogous to it?
Conjugate symmetry_ I only know about a conjugate in a group which says $b$ is said to be conjugate of $a$ if there exist $x$ such that $b=xax^{-1}$
I mean there has to be someone who goes an extra mile to make sense of the nonsense and define and generalize every possible term. If not then what's the problem they had faced until now.
I am just a 3rd year undergraduate student with interest in math. Please, don't assume that I know a lot of math and try to use simple terms. Thank You.