How many vectors are needed to compute (p,q)-summing norms?

We will show that for $q<p$ there exists an $\al < \infty$ such that \[ π_{pq}(T) \pl \le c_{pq} π_{pq}^{[n^α]}(T) \mbox{for all $T$ of rank $n$.}\] Such a polynomial number is only possible if $q=2$ or $q<p$. Furthermore, the growth rate is linear if $q=2$ or $\frac{1}{q}-\frac{1}{p}>\frac{1}{2}$. Unless $\frac{1}{q}-\frac{1}{p}=\frac{1}{2}$ this is also a necessary condition .

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