Let $\ell^\infty$ be the space of bounded sequences with the maximum norm and $c_0$ the space of sequences that have limit $0$ with the same norm. I use the notation $E^*$ for the dual of a normed space $E$.
I know that $c_0$ is a closed subspace of $\ell^\infty$, that $c_0^{**}=\ell^\infty$, and that both $c_0$ and $\ell^\infty$ are not reflexive. How can I prove that $\ell^\infty/c_0$ is not reflexive?
My best guess has been using that a space is reflexive iff its dual is reflexive. The dual of the quotient is $(\ell^\infty/c_0)^*=c_0^\perp$, the orthogonal of $c_0$ in the dual of $\ell^\infty$ (thanks to David for the correction). This is a closed subspace of $(\ell^\infty)^*$ and $(\ell^\infty)^*$ is also not reflexive, but I don't think this implies that $c_0^\perp$ is not reflexive. I also tried using the characterization "A Banach space is reflexive iff the unit ball is weakly compact" but I haven't been able to prove that the ball isn't weakly compact.
