prove that $T''$ is not injective (difficult computation)

Question

Let $T:c_0 \to c_0$ defined by $T(\{s_j\}_j)=\{s_{j+1}-s_j\}_j$. Prove that $T''$ is not injective.

I tried even knowing that $(c_0)' \sim l^1$,in fact if $F\in (c_0)'$ then there exist $s=(s_j)\in l^1$ such that $F=F_s$ and $F_s(t_j)=\sum s_jt_j$ , and similar for $(l^1)' \sim l^{\infty}$.

It's easy to prove that $T''$ is injective if and only iff the image of $T'$ is dense on $X'$. Maybe this is easy to prove.

Answer 1

Hint: for $s \in \ell^\infty$, $(T'' s)_j = s_{j+1} - s_j$ as well. So what $s$ would make those all $0$?