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We prove that, given a constant $K> 2$ and a bounded linear operator $T$ froma JB$^*$-triple $E$ into a complex Hilbert space $H$, there exists a norm-onefunctional $\psi\in E^*$ satisfying $$\|T(x)\| \leq K \, \|T\| \,\|x\|_{\psi},$$ for all $x\in E$. Applying this result we show that, given $G >8 (1+2\sqrt{3})$ and a bounded bilinear form $V$ on the Cartesian product oftwo JB$^*$-triples $E$ and $B$, there exist norm-one functionals $\varphi\inE^{*}$ and $\psi\in B^{*}$ satisfying $$|V(x,y)| \leq G \ \|V\| \,\|x\|_{\varphi} \, \|y\|_{\psi}$$ for all $(x,y)\in E \times B$. These resultsprove a conjecture pursued during almost twenty years.

Grothendieck's inequalities for JB$^*$-triples: Proof of the Barton-Friedman conjecture