Let $A:X\to Y$ be a continuous linear map.
Let $A^*:Y^*\to X^*$ denote the transpose of $A$ (also referred to as the dual transformation of $A$)
Let $\ker A ^⊥$ denote the annihilator of $\ker A$.
1) Prove that $\overline{\text{im} A^*}\subset\ker A ^⊥$
2) Setting $X=Y=l_1$ and $A:(u_n)\to \left(\frac{u_n}{n}\right)$, prove that $\overline{\text{im} A^*}\subsetneq\ker A ^⊥$
While the first question is very easy to deal with using the very definitions of $A^*$ and $\ker A ^⊥$, I'm stuck with the second question.
Since $\ker A = {0}$, we have $\ker A ^⊥=l_1^*$, so it boils down to finding some $f\in l_1^*\simeq l_\infty$ such that $f\notin \overline{\text{im} A^*}$.
I can't really picture what ${\text{im} A^*}$ is, so that makes finding the convenient $f$ really hard for me.