This is a homework question, so I am looking for a hint rather than an answer.
If $S$ is finite, then $\mathbb{F}^S$ is a finite dimensional vector space. (Note: $\mathbb{F}^S$ is the set of functions from $S$ to $\mathbb{F}$.
My proof goes like so:
1) $\mathbb{F}^S$ is a finite dimensional vector space if a list of vectors $f_1,f_2,\dots,f_n$ in $\mathbb{F}^S$ spans $\mathbb{F}^S$.
2) The list $f_1,f_2,\dots,f_n$ of vectors from $\mathbb{F}^S$ spans $\mathbb{F}^S$ if span$(f_1,f_2,\dots,f_n)=\mathbb{F}^S$.
Choose an arbitrary element $f$ from span$(f_1,f_2,\dots,f_n)$. Then we can write $f$ as
$$f=a_1f_1+a_2f_2+\cdots+a_nf_n,$$ where each of the $a_i$ belongs to $\mathbb{F}$. Since $\mathbb{F}^S$ is a vector space, then each element of the sum is also a vector belonging to $\mathbb{F}^S$ and thus so is the sum itself. So $f$ itself belongs to $\mathbb{F}^S$.
My problem: I never used the finiteness of $S$, so I have a feeling that this proof is flawed. Can somebody give me a hint as to where I am lacking or where I need to use the finiteness of $S$? Thanks.