An operator is injective if and only if the range of its adjoint separates points Let $T \in L(E,F)$ where $E,F$ are normed spaces.  Then, $T$ is injective if and only if $ T^*(F') \subset E' $  separates points of $E$. 
$T^*$ means adjoint of T.
I do not have a starting point. I don't even know what separates means in this problem. It should have been clearer , but this is the way it is written. Can someone give a definition of separation in this context and point me to a direction after that?

It's convenient to use the bracket notation $\langle x,\varphi \rangle =\varphi(x)$ when discussing the spaces and their duals. So, the adjoint $T^*$ satisfies $\langle Tx, \varphi\rangle = \langle x, T^*\varphi\rangle$ for all $x\in E$ and $\varphi\in F^*$. Hence
$$
\|Tx\| = \sup_{\|\varphi\|_{F^*}=1} |\langle Tx, \varphi\rangle |
= \sup_{\|\varphi\|_{F^*}=1} |\langle x, T^*\varphi\rangle|
$$
which implies 
$$\ker T=\bigcap_{\|\varphi\|_{F^*}=1} \ker T^*\varphi
= \bigcap_{\psi\in \operatorname{ran} T^*} \ker \psi
$$ 
Both directions of the claim follow at once.
