Does there exist an equivalent Frechet differentiable norm on $\ell_1$ and $c_0$? I think we can not find an equivalent norm on $\ell_1$ but we could find an equivalent norm on $c_0$, I do not prove this claim, only I guess it is true, please help me. thanks

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    $\begingroup$ Since $\ell_1=c_0^*$, having a Frechet differentiable norm on $\ell_1$ would imply that $c_0$ is reflexive, which it is not. $\endgroup$ – user147263 Jun 14 '14 at 5:43

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