  In this paper we survey some recent results concerning the numerical index
$n(\cdot)$ for large classes of Banach spaces, including vector valued
$\ell_p$-spaces and $\ell_p$-sums of Banach spaces where $1\leq p < \infty$. In
particular by defining two conditions on a norm of a Banach space $X$, namely a
Local Characterization Condition (LCC) and a Global Characterization Condition
(GCC), we are able to show that if a norm on $X$ satisfies the (LCC), then
$n(X) = \displaystyle\lim_m n(X_m).$ For the case in which $ \mathbb{N}$ is
replaced by a directed, infinite set $S$, we will prove an analogous result for
$X$ satisfying the (GCC). Our approach is motivated by the fact that $
n(L_p(\mu, X))= n(\ell_p(X)) = \displaystyle \lim_m n(\ell_p^m (X))$ \cite
{aga-ed-kham}.
