An Isomorphic Version of Nakano’s Characterisation of C0(Σ)

In 1941 Nakano gave a characterisation of real spaces C₀(Σ) as M-spaces satisfying an additional restriction on their norm. We give an isomorphic version of Nakano’s result which involves showing that for any Banach lattice (a suitably modified version of) Nakano’s condition is equivalent to the norm on the Dedekind completion being Fatou.