In Axler's book on Linear Algebra he writes ($\mathbb{F}$ is here either $\mathbb{R}$ or $\mathbb{C}$):
The scalar multiplication in a vector space depends upon $\mathbb{F}$. Thus when we need to be precise, we will say that $V$ is a vector space over $\mathbb{F}$ instead of saying simply that $V$ is a vector space. For example, $\mathbb{R}^n$ is a vector space over $\mathbb{R}$, and $\mathbb{C}^n$ is a vector space over $\mathbb{C}$.
What is actually meant by $\mathbb{R}^n$ being a vector space over $\mathbb{R}$ and how can one verify that it is? What are all the implications of this? Is it just that any scalar multiplication in $\mathbb{R}^n$ depends upon numbers in $\mathbb{R}$?
To me, $\mathbb{R}^n$ feels larger than $\mathbb{R}$, so I find the wording, that one is over the other, hard to digest...