  We measured the spin asymmetry in the scattering of 100 GeV
longitudinally-polarized muons on transversely polarized protons. The asymmetry
was found to be compatible with zero in the kinematic range $0.006<x<0.6$,
$1<Q^2<30\,~\mbox{GeV}^2$. {}From this result we derive the upper limits for
the virtual photon--proton asymmetry $A_2$, and for the spin structure function
$g_2$. For $x<0.15$, $A_2$ is significantly smaller than its positivity limit
$\sqrt{R}$.
