Let $\displaystyle F$ be a field and $\displaystyle X$ be a non-empty set. Let $\displaystyle F^*$ be the set of all functions $\displaystyle f:X\rightarrow F$ together with addition, defined by

$\displaystyle (f_1+f_2)(x)=f_1(x)+f_2(x)$, $\displaystyle \forall f_1,f_2\in F$ and all $\displaystyle x\in X$,

and scalar multiplication defined by

$\displaystyle (cf)(x)=cf(x)$, $\displaystyle \forall f\in F^*$ and $\displaystyle c\in F$.

Prove that $\displaystyle F^*$ is a vector space over $\displaystyle F$ by verifying each axiom in the definition.

You will need to refer to the axioms of the field $\displaystyle F$. It is important to make a distinction between a function $\displaystyle f\in F^*$ and one of its values $\displaystyle f(x)\in F$.

The main issue I'm having with this one is wording, and the fact that I just got back into this after the summer, so I'm virtually brain-dead until I can get settled in again. The fact that I only just started learning anything about algebra-related fields doesn't help. Any help on this would be appreciated.