Limit Theorems for Numerical Index

We improve upon on a limit theorem for numerical index for large classes of Banach spaces including vector valued $\ell_p$-spaces and $\ell_p$-sums of Banach spaces where $1\leq p \leq \infty$. We first prove $ n_1(X) = \displaystyle \lim_m n_1(X_m)$ for a modified numerical index $n_1(\, .\,)$. Later, we establish if a norm on $X$ satisfies the local characterization condition, then $n(X) = \displaystyle\lim_m n(X_m).$ We also present an example of a Banach space where the local characterization condition is satisfied.