From wikipedia it says about coordinate space
By adding further structure and constraints, a coordinate space may be used to construct an object such as a vector space or a manifold.
I have not found anywhere how to construct a vector space $V$ starting from a coordinate space.
Can you provide me an example about this construction ? Cartesian coordinate system should be fine.
Since you have a coordinate space, you naturally have a set of the form $S=\prod\limits_{i=1}^n C_i$. In order to extend this set to a vector space all you need is to choose an underlying $\textit{field}$, $\mathbb{F}$ for the vector space and define the maps
$$ +: S \times S \to S $$ $$ .: S \times \mathbb{F} \to S $$
such that $(S,+)$ forms an additive abelian group and $.$ is distributive over scalar addition and field addition, existence of scalar inverse.
A simple example would be to look at $S= \mathbb{R}^n$ and choose $\mathbb{F}=\mathbb{R}$. For any elements,
$$x=(x_1,x_2,\dots.x_n)$$ $$y=(y_1,y_2,\dots.y_n)$$
in $\mathbb{R}^n$ and for any $\alpha \in \mathbb{F}$ simply define,
$$x+y= \sum\limits_{i=1}^n(x_i+y_i)$$ $$\alpha x=(\alpha x_1, \alpha x_2, \dots, \alpha x_n)$$.
Then you can easily verify that the above choice satisfies the condition for a vector space.