The following is extracted from the book 'Lipschitz Algebras' by N. Weaver. (page $36$)
It is standard that a bounded linear functional $\phi \in Lip_0(X)^*$ belongs to the predual of $Lip_0(X)$ if and only if it is continuous with respect to the weak$^*$ topology.
How to prove the above statement?
Note that $X$ is a pointed metric space (We denote this point as $0$). $Lip_0(X)$ is the set of all real-valued Lipschitz functions which satisfy $f(0)=0$.