Let $V$ and $W$ be vector spaces over a field $K$. If a linear map $L:V \rightarrow W$ is surjective then its dual is injective. If $V$ and $W$ are finite dimensional then the converse holds, i.e. $L^*:W^* \rightarrow V^*$ injective implies $L$ surjective.
I have proved both statements but I don't see where I used the finite dimensional requirement for the second. Here is my proof:
Assume $L$ is not surjective, say the element $e_i$ of the basis of $W$ is not in the image of $L$. Take its corresponding dual $\alpha_i \in W^*$, then $L^*(\alpha_i)=\alpha_i \circ L =0$ so the kernel of $L^*$ is not 0 and therefore $L^*$ is not injective.