I guess this is probably asked before but I can not find it.
Let $X$ be a Banach space, and let $\ell x = 0$ for all $\ell \in X'$. Then $x = 0$
If all projections $\pi_\alpha x = 0$ and hence get "all coordinates" equal zero. But that probably not hold and I read you need Hahn-banach to prove it.