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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.

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

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