# Only zero in kernel of every element of a subspace of the dual does not imply density of said subspace for non-reflexive Banach spaces

Let X be a Banach space and $$M \subset X'$$ a subspace of its dual space.

If X is reflexive we know the following statements are equivalent:

(i) $$M$$ is dense in $$X'$$

(ii) $$x \in X$$ and $$\phi(x)=0$$ for all $$\phi \in M$$ implies $$x=0$$

Now (ii) $$\rightarrow$$ (i) is not true if X is non-reflexive.

I struggle however to come up with a counterexample.

• Consider $c_0$ living in (the non-separable) $X'=\ell_\infty$ ($X=\ell_1$). – David Mitra May 24 at 13:42