$\ell x = 0$ 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.

I think you have the right intuition, namely, if $x\neq 0$ you want to construct some functional $\ell\in X'$ such that $\ell x\neq 0$. The Hahn-Banach theorem makes this pretty easy to do. 
Assume $x\in X$ is nonzero. Let $Y\subseteq X$ be the $1$-dimensional subspace spanned by $x$. Then define a linear map $L\colon Y\to \mathbb{R}$ by $L(\lambda x) = \lambda$. Here I'm assuming you're working over $\mathbb{R}$, but just replace $\mathbb{R}$ with $\mathbb{C}$ if you're working over $\mathbb{C}$. This $L$ is a continuous linear functional on $Y$, and $L(x) = 1$. The Hahn-Banach theorem tells us that $L$ extends to a continuous linear function $\ell\colon X\to \mathbb{R}$. This gives an $\ell\in X'$ such that $\ell(x) = 1$.
