Definition:Real Vector Space

Theorem
Let $$\mathbb{R}$$ be the set of real numbers.

Then the $\mathbb{R}$-module $$\mathbb{R}^n$$ is a vector space.

It follows directly, by setting $$n=1$$, that the $\mathbb{R}$-module $$\mathbb{R}$$ itself can also be regarded as a vector space.

Comment
The $\mathbb{R}$-vector space $$\mathbb{R}^3$$ can be shown (given appropriate assumptions about the nature of the universe) to be isomorphic to the spatial universe.

Similarly, the $\mathbb{R}$-vector space $$\mathbb{R}^2$$ can be shown to be isomorphic to an infinite flat plane, and the $\mathbb{R}$-vector space $$\mathbb{R}$$ to an infinite straight line.

Thus $$\mathbb{R}, \mathbb{R}^2$$ and $$\mathbb{R}^3$$ have a particular and obvious significance.