 To prove the inequality above set

with |"a""ij"| ≤ 1 chosen so that

Then

Hence

Taking 2"m"th roots and letting "m" tend to ∞,

which immediately implies the inequality.There is a generalization of the Cotlar–Stein lemma with sums replaced by integrals.Let "X" be a locally compact space and μ a Borel measure on "X".Let "T"("x") be a map from "X" into bounded operators from "E" to "F" which is uniformly bounded and continuous in the strong operator topology.