= 2-summing operator
{title2=$\pi_2(T)$}
A <bounded linear operator> $T:E\to F$ is 2-summing if $\sum_j\|Tx_j\|^2\le C^2\sup_{\phi\in B_{E^*}}\sum_j|\phi(x_j)|^2$ for every finite family; $\pi_2(T)$ is the least $C$. For operators between <Hilbert spaces>, this is equivalent to being a <Hilbert-Schmidt operator>, with $\pi_2(T)=\|T\|_{\mathrm{HS}}$. The upper bound follows by applying the <Hilbert-Schmidt norm> ideal inequality to the column operator of the family; testing finite orthonormal families gives the reverse bound.
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