Show that a subspace of $c_{0}$ with the norm induced by $c_{0}$, can’t be isomorphic to $l_1$.

Question

Show that a subspace of $c_{0}$ with the norm induced by $c_{0}$, can’t be isomorphic to $l_1$.

Any idea or hint? I think have to show that the given subspace has a certain property that $l_1$ no has (or the reverse) but I can’t understand which property... thank you!

Answer 1

The dual space of $\ell^1$ is isometrically isomorphic to $\ell^\infty,$ which is nonseparable.

However, the dual of $c_0$ is isometrically isomorphic to $\ell^1,$ which is separable. Hence the dual of any subspace of $c_0$ is separable as well.

This argument actually shows that there is not a linear homeomorphism between $\ell^1$ and a subspace of $c_0$, let alone an isometric isomorphism.