Let $V$ be a finite dimensional vector space over a finite field $k$.
Denote by $\widehat{V}$ the group of additive characters $V \to \mathbb{C}^*$ and $V^*=Hom_k(V,k)$ the linear dual. Fix a non-trivial additive character of the field $\chi : k \to \mathbb{C}^*$. Consider the following map:
$$V^* \to \widehat{V},\alpha \mapsto \{v \mapsto \chi(\alpha(v))\}$$
I'd like to show that this is an isomorphism. I had no problem showing it was injective. To show surjectivity I used my knowledge of the charater of $k$ to write everything explicitly and it worked in the end. I think this should be a lot simpler though. In particular I think there should be a proof that works for continuous characters with $k$ local field. So:
What's the most natural way to show that this is a surjection? (using the least amount of explicit calculations).
The group of additive characters of a finite abelian group is of cardinality equal to that of the group you started with. Likewise $V^*$ has cardinality equal to that of $V$. So for any function $V^* \longrightarrow \widehat{V}$, injectivity and surjectivity are equivalent.