$\ell^{1}$ not reflexive implies $\ell^{\infty}$ not reflexive

Question

I knew that the dual space of $\ell^1$, a separable space is not reflexive because its dual space is $\ell^\infty$, which is not separable. But, I do not know whether $\ell^\infty$ is reflexive or not because I was unable to use the result on the relation between separability and reflexivity to prove it. Is there any hint on this?

Answer 1

Here is an answer expanding on the approach indicated in the comments.

We assume the Banach space $\ell^{1}$ is not reflexive. Then by this result the dual $(\ell^{1})^{*}$ is not reflexive. Since $(\ell^{1})^{*}$ is (isometrically) isomorphic to $\ell^{\infty}$ by this result and isomorphisms preserve reflexivity by this result, it follows that $\ell^{\infty}$ is not reflexive.