TITLE: $\ell x = 0$ for all $\ell \in X'$ for banach spaces
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: An interesting addendum.  For general Banach space, the Hahn-Banach theorem, or some other consequene of the Axiom of Choice, is required to prove this.  In plain ZF set theory it cannot be proved.  A nice example:  
$X = l^\infty/c_0$, a fairly concrete Banach space.  Certainly $X$ has nonzero elements (for example, the equivalence class of $(1,1,1,\dots)$).  But (in only ZF) one cannot show that $X$ has any nonzero linear functionals at all.