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.


About mlbaker

just another guy trying to make the diagrams commute.
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