This question already has an answer here:
Given an arbitrary nonzero vector space $V$, is there a nonzero linear functional on $V$, without assuming axiom of choice? I know that by assuming existence of a basis for $V$, we can consider the dual basis for a subspace of $V^*$, which justifies the existence of nonzero linear functional, but this argument fails without axiom of choice. I guess there may not always be a nonzero linear functional, but my knowledge on axiom of choice and infinite-dimensional vector spaces is lacking. A quick search on Google fails to give an answer.
Note that I am not talking about normed spaces or continuous linear functionals, just plain vector spaces with no additional structure.