Non‐linear factorization of linear operators

We show, in particular, that a linear operator between finite‐dimensional normed spaces, which factors through a third Banach space Z via Lipschitz maps, factors linearly through the identity from L ∞ ([0, 1], Z ) to L 1 ([0, 1], Z ) (and thus, in particular, through each L p ( Z ), for 1 ⩽ p ⩽ ∞) with the same factorization constant. It follows that, for each 1 ⩽ p ⩽ ∞, the class of ℒ p spaces is closed under uniform (and even coarse) equivalences. The case p = 1 is new and solves a problem raised by Heinrich and Mankiewicz in 1982. The proof is based on a simple local–global linearization idea.