$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla f\|_{\infty}$. Following from Some natural subspaces and quotient spaces of $L^1$, $\big(\Lip_0(\mathbb R^d), \|\cdot\|\big)$ is a Banach space. Now we endow $\Lip_0(\mathbb R^d)$ with an alternative topology, denoted by $w$ and generated by the open sets $\mathcal O_{u}(f;\epsilon)$ as below:

$$\mathcal O_{u}(f;\epsilon) \quad:=\quad \left\{g\in \Lip_0(\mathbb R^d):~ \left|\int_{\mathbb R^d} \big[\nabla(f-g)(x)\cdot u(x)\big]\right| dx <\epsilon \right\},$$

where $f\in \Lip_0(\mathbb R^d)$, $u\in L^1(\mathbb R^d;\mathbb R^d)$ and $\epsilon>0$. My question is as follows: Let $(f_{\lambda})_{\lambda\in\Lambda}\subset \Lip_0(\mathbb R^d)$ be a net $w$-converging to $f\in \Lip_0(\mathbb R^d)$. **Could we select a subnet $(f_{\lambda_{\alpha}})_{\alpha}$ s.t. $\sup_{\alpha}\|f_{\lambda_{\alpha}}\|<\infty$?**

Lipschitz Algebras, second edition. $\endgroup$