If $K$ is a field, the dual of $K^{({\mathbb N})}$ is $K^{\mathbb N}$, and axiom of choice implies that the natural map from $K^{({\mathbb N})}$ to the dual of $K^{\mathbb N}$ is far from being surjective. Is there any field for which one can prove this without using AC?

(A related result that I found surprising is that ${\rm Hom}({\mathbb Z}^{({\mathbb N})},{\mathbb Z})={\mathbb Z}^{\mathbb N}$ and ${\rm Hom}({\mathbb Z}^{\mathbb N},{\mathbb Z})={\mathbb Z}^{({\mathbb N})}$.)