Limit Theorems for Numerical Index

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