Published online by Cambridge University Press: 08 April 2021
Let
$\mathrm {Lip}_0(M)$
be the space of Lipschitz functions on a complete metric space M that vanish at a base point. We prove that every normal functional in
${\mathrm {Lip}_0(M)}^*$
is weak* continuous; that is, in order to verify weak* continuity it suffices to do so for bounded monotone nets of Lipschitz functions. This solves a problem posed by N. Weaver. As an auxiliary result, we show that the series decomposition developed by N. J. Kalton for functionals in the predual of
$\mathrm {Lip}_0(M)$
can be partially extended to
${\mathrm {Lip}_0(M)}^*$
.