Normal functionals on Lipschitz spaces are weak* continuous

[EN] Let Lip0(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 Lip0(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 Lip0(M) can be partially extended to Lip0(M)¿.