$\ell x = 0$ for all $\ell \in X'$ for banach spaces 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.

Hahn-Banach theorem says that if $X$ is any normed space then there exists $l\in X'$ such that $lx= \|x\|$ and $|lx|\leq \|x\|$ for all $x\in X$.[Rudin Corollary at page 59].
Now if $x\neq 0$, then we can not have $l(x)=0$ for all $l\in X'$. 
