In any Banach space $B$, if $S$ is a  countable set of bounded linear maps, there is $a \in B$ such that $T \in S \mapsto Ta \in B$ is injective.
This follows easily from the the Baire Category Theorem: $B$ is not the union of the countably many closed, nowhere dense sets $\{x \in B \mid T_i x = T_j x\}$.