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…
Best Answer
When you give the axioms of a vector space $V$, you have to talk about a certain field: for example, in the axioms that talk about scalar multiplication, you'll say something like "for every $x \in \mathbb{F}$ and every $v \in V$ there's an element $xv$ of $V$....". In this case we say that "$V$ is a vector space over $\mathbb{F}$".
So you're right: $V$ is a vector space over $\mathbb{F}$ means that the scalars you can multiply by are elements of $\mathbb{F}$. It's just a conventional use of the word over.
The reason to mention it is that sometimes which field you are working over is not obvious. Take $\mathbb{C}^n$. It is a vector space over $\mathbb{C}$ in the obvious way, but it is also a vector space over $\mathbb{R}$ (if you can use elements of $\mathbb{C}$ as scalars then you can certainly use $\mathbb{R}$ as $\mathbb{R} \subset \mathbb{C}$). What's more it has dimension $n$ over $\mathbb{C}$ but $2n$ over $\mathbb{R}$ so it really matters which field you work over.