Limit theorems for the numerical index

We improve on a limit theorem (see Martin et al. (2011) [13], Th. 5.1) for numerical index n ( ⋅ ) for large classes of Banach spaces including vector valued ℓ p -spaces and ℓ p -sums of Banach spaces where 1 ≤ p < ∞ . We introduce two conditions on a Banach space X , a local characterization condition (LCC) and a global characterization condition (GCC). We prove that if a norm on X satisfies the (LCC), then n ( X ) = lim m n ( X m ) . An analogous result, in which N will be replaced by a directed, infinite set S will be proved for X satisfying the (GCC). We also present examples of Banach spaces satisfying the above mentioned conditions.