Let $F$ be a infinite dimensional Banach space and $T\subseteq S\subseteq F^*$ such that $\mbox{dim}(S/T)=1$. If $T$ is not weak*-dense in $F^*$, then $S$ can be weak*-dense in $F^*$?
I've been searching for basic results to decide and prove, but I couldn't make a good progress. Then, I'd be grateful if someone could give me any hits. Thanks in advance!
Fix $x_0\in F$ with $x_0\ne0$. Let $T=\{\lambda\in F^*:\lambda x_0=0\}$ and $S=F^*$.