Preduals and $c_0$ I know that $c_0$ does not have a predual, but if we put an equivalent norm on $c_0$, can this space have a predual?

No. If the norm is equivalent any bounded sequence in the original space would still be bounded. Now use Banach-Alaoglu. Take the sequence $s_N = \chi_{[0,N]}$. That sequence must have a weak-$\ast$ accumulation point. But the predual $E$ is an isometric subset of $c_0^\ast = \ell^1$ and that sequence satisfies that $\langle \psi, s_N \rangle \to \langle \psi, \chi_{\mathbb{N}} \rangle$ for every $\psi \in \ell_1$ and thus for every $\psi \in E$. But $\chi_{\mathbb{N}} \not\in c_0$.
