metrics $d_{\infty,i}$ and $d_{1,i}$ on $M_i$, with the property, that a given space $X$ is non reflexive if and only if there are injections $\Phi_i: M_i\to X$ and $0<c\le 1$ so that $cd_{\infty,i}(x,y)\le \|x-y\|\le d_{1,i}(x,y)$, for all $x,y\in M_i$”. In Section \[S:13\] we will discuss the difficulties, in arriving to a characterization of reflexivity of the first form. Nevertheless, if we restrict ourselves to the class of reflexive spaces we arrive to a metric characterization for the [*complexity*]{} of a given space, which we measure by the Szlenk index, using test spaces. Roughly speaking, the higher the Szlenk index is of a given Banach space, the more averages of a given weakly null sequence one has to take to arrive to a norm null sequence. For a precise formulation of this statement we refer to Theorem \[T:6.3\]. For the class of separable and reflexive spaces we will introduce an uncountable family of metric spaces $(M_\alpha)_{\alpha<\omega_1}$ for which we will show that the higher the complexity of a given reflexive and separable space $X$ or its dual $X^*$ is, the more members of $(M_\alpha)_{\alpha<\omega_1}$ can be uniformly bi-Lipschitzly embedded into $X$.

The statements of our main results are as follows. The 