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.

Proof
From the definition, a vector space is a unitary module whose scalar ring is a division ring.

As $\mathbb{R}$ is a field, $$\mathbb{R}$$ is a division ring.

So the $\mathbb{R}$-module $\mathbb{R}^n$ fits the description.

Comment
The real vector spaces have direct applications to the real world. In fact, it could be suggested that they are the interface between mathematics and physical reality, as follows:


 * From the definition of the Real Number Line, the $\mathbb{R}$-vector space $$\mathbb{R}$$ is isomorphic to $$\mathbb{R}$$ to an infinite straight line.


 * From the definition of the Real Number Plane, the $\mathbb{R}$-vector space $$\mathbb{R}^2$$ is isomorphic to $$\mathbb{R}$$ to an infinite flat plane.


 * 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.