Structure of space

Question

Can we consider the structure of space( only space not space-time ) to be that of a vector space? Why can we or why can't. And why cant we give a vector space structure to space-time?

Answer 1

A vectors space has a preferred vector, the zero vector. Instead there is no preferred point in classical (flat) physical space. Physical space is better described by a so-called three-dimensional affine space. Metric tools are then represented by a scalar product in the space of translations. This space which describes rigid movements of the points in the affine space is a vector space, but it is not the physical space itself, just describes rigid movements in physical space.

Similarly, the spacetime of special relativity is well described by a four-dimensional affine space whose space of translations is equipped with a Minkowskian scalar product.