Let $(E;\Vert \cdot \Vert )$ be a normed vector space. Show that given $x, y \in E$, $x \neq y$, there exists $f \in E'$ ($E'$ is the bounded dual space of $E$) such that $f(x) \neq f(y)$.
I started trying to show that there always is an injective map on the dual, which would imply this result, but it happens to be case that some bounded dual spaces do not have any injective maps.
Any help would be greatly appreciated. Thanks