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.

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About mlbaker

just another guy trying to make the diagrams commute.
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