In this paper we have introduced a new concept on the convergence of a sequence of the nonlinear Lipschitz (Lip-) functionals, which would be called an L*-convergence, and we have considered its applications in Banach spaces. This convergence is very similar to the weak* (W*-) convergence of the sequence of the bounded linear functionals, but there are some differences. By the L*-convergence, we have considered the problem on the relations of the compactness between the Lip-operator and its Lip-dual operator, and we have obtained the mean ergodic theorems for the Lip-operator.
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