We show that the numerical index of a c 0-, l 1-, or l ∞-sum of Banach spaces is the infimum numerical index of the summands. Moreover, we prove that the spaces C(K, X) and L 1 (µ, X) (K any compact Hausdorff space, µ any positive measure) have the same numerical index as the Banach space X. We also observe that these spaces have the so-called Daugavet property whenever X has the Daugavet property.

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We show that an infinite-dimensional real Banach space with numerical index 1 satisfying the RadonNikodỳm property contains l 1 . It follows that a reflexive or quasi-reflexive real Banach space cannot be re-normed to have numerical index 1, unless it is finite-dimensional.

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