Dual Space isomorphic with all function

Question

Let $\mathbb X$ be an infinite set and $k$ be a field
Define $$C_c(\mathbb X,k)=\{f\in k^\mathbb X:f(x)=0\text{ for all but finite }x's\}$$

Prove $C_c(\mathbb X,k)^*\cong k^\mathbb X$

If it were finite dimensional, it would obviously not be true due to dimensional difference.

I know $\phi \in C_c(\mathbb X,k)^*$ maps $f(\in C_c(\mathbb X,k))$ to a scaler (belonging to k?)

I don't think I have anymore clue. Please give hints. This is supposed to be homework

Answer 1

Note that $C_c(\mathbb{X},k)$ is the the $k$-vector space spanned by $\{1_x: x\in X\}$. So an element $\phi$ of the dual vector space $C_c(\mathbb{X},k)^\vee$ determines a function $f\colon\mathbb{X}\to k$, namely $x\mapsto\phi(1_x)$, and conversely any choice of function $f\colon\mathbb{X}\to k$ determines $\phi$. Now check that this bijection is actually a $k$-linear map.