  We prove that, given a constant $K> 2$ and a bounded linear operator $T$ from
a JB$^*$-triple $E$ into a complex Hilbert space $H$, there exists a norm-one
functional $\psi\in E^*$ 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 of
two JB$^*$-triples $E$ and $B$, there exist norm-one functionals $\varphi\in
E^{*}$ and $\psi\in B^{*}$ satisfying $$|V(x,y)| \leq G \ \|V\| \,
\|x\|_{\varphi} \, \|y\|_{\psi}$$ for all $(x,y)\in E \times B$. These results
prove a conjecture pursued during almost twenty years.
