Grothendieck theorem for L1 to Hilbert operators
= Grothendieck theorem for L1 to Hilbert operators
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Every <bounded linear operator> $T:L^1(\mu)\to H$ into a <Hilbert space> is an <absolutely summing operator>, with $\pi_1(T)\le K_G\|T\|$. 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.