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Published online by Cambridge University Press: 24 October 2008
A simple proof is given that if X is a normed vector lattice, 1 ≼ p ≼ ∞, and the norm in X is p-additive, then the norm is p-superadditive if p < ∞. If p = ∞, then X satisfies Kakutani's classical M-space condition. The proof uses duality but is otherwise elementary.