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