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.
The answer is negative. In other words, there are models of set theory (without the axiom of choice) for which there are vector spaces $V\neq\{0\}$ such that $V^*=\{0\}$. See here, for instance.