= Pietsch factorization theorem
{c}
For an <absolutely p-summing operator> $T:E\to F$, there is a <probability measure> $\mu$ on the dual unit ball with its <weak-star topology> such that $\|Tx\|\le\pi_p(T)(\int|\phi(x)|^p\,d\mu(\phi))^{1/p}$. Thus $T$ factors through the closed span of the evaluation functions in $L^p(\mu)$, with the final operator having norm at most $\pi_p(T)$. Conversely this factorization implies the summing inequality by summing under the integral. Existence of $\mu$ follows from finite-dimensional <Hahn-Banach separation theorem>: failure for finitely many vectors would contradict the summing inequality for a nonnegative weighted family. Compactness of probability measures then combines the finite constraints. The final operator need only be defined on the indicated closed subspace, not on all of $L^p(\mu)$.
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