## Preduals of Banach spaces

I have a slight addendum to my recent post on PMATH 753: in A1P3, we showed that $\pmb{c}(\mathbb{Z})^* \cong \ell_1(\mathbb{Z} \cup \{ -\infty, +\infty \})$. However, it is clear that adding these two extra points is unnecessary (the resulting set is still countable); we also have $\pmb{c}(\mathbb{N})^* \cong \ell_1(\mathbb{N})$ as well as $\pmb{c}_0(\mathbb{N})^* \cong \ell_1(\mathbb{N})$. However in the first bonus problem, A1P3d, we showed that there is no linear isometric isomorphism $\pmb{c} \cong \pmb{c}_0$. This shows an odd phenomenon (pointed out to me by Juno) that preduals of Banach spaces need not be unique.