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.

If the closed span of the Rademacher functions were a complemented subspace of $L^1(0,1)$, compose a projection with the inverse of its l2 sequence space parametrization. The Grothendieck theorem for L1 to Hilbert operators would make this map an absolutely summing operator. Its composition with the parametrization would make the identity on the infinite-dimensional l2 sequence space absolutely summing, contradicting its coordinate-vector test.