Consider the Banach space $X$ of Lipschitz functions $g:[0,1] \to \mathbb{R}$ such that $g(0)=0$, with the norm of $g$ given by its Lipschitz constant, i.e. $||g||_L = \sup_{x \neq y } \frac{g(x)-g(y)}{|x-y|}$.

Suppose that for some constant $K$ we have a bounded sequence of functions $\{f_n\} \subseteq X$ such that $||f_n||_L \leq K$ for all $n$. By the Arzelà–Ascoli theorem, there is a subsequence which converges (in the $C^0$ norm) to a continuous function $f:[0,1] \to \mathbb{R}$. However, this convergence is not necessarily in the $||\cdot ||_L$ norm. Nevertheless, the function $f$ is Lipschitz and $||f||_L \leq K$.

When could one expect the subsequence of $f_n$ to converge weakly to the function $f$?