 Grothendieck theorem for L1 to Hilbert operators

Every bounded linear operator $T:L^1(\mu )\to H$ into a Hilbert space is an absolutely summing operator, with ${\pi }_1(T)\le K_G\Vert T\Vert$. For simple functions on a common disjoint partition, apply the Grothendieck inequality to the vectors $T(1_A/\mu (A))$ and norming vectors of the images. The scalar sign or phase supremum is the weak 1-norm of the family. Approximation by simple functions proves the general result.