Can I use the mapping notation to describe this vector space? So, I recently learned that the statement $v \in \mathbf F^S$, where $\mathbf F$ is a field and $S$ is a set, can be rewritten as  $v : S \rightarrow \mathbf F$.
Does this imply that $ x\in \mathbf F^n$ can be rewritten as $x: n\rightarrow \mathbf F$?
$x$ is a mapping of $ n $ onto $ \mathbf F$?

As @peek-a-boo pointed out,  there's a way to.  The caveat is that $n$ refers to an $n$ element set.
Then the vector space $\Bbb F^n$ can then be considered as the space of functions from $\{1,2,\dots,n\}$ to $\Bbb F$.
This is a sort of functional analysis approach.
