$\DeclareMathOperator\Lip{Lip}$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 $\|f\|:=\|\nabla f\|_{\infty}$. Then $\big(\Lip_0(\mathbb R^d), \|\cdot\|\big)$ 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 $\mathcal O_{u}(f;\varepsilon)$ as below:

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

where $f\in \Lip_0(\mathbb R^d)$, $u\in L^1(\mathbb R^d;\mathbb R^d)$ and $\varepsilon>0$. Let $\mathcal C:=\{f\in\Lip_0(\mathbb R^d): \|f\|\le 1\}$ and denote by $\overline {\mathcal C}$ its $w$-closure. Could we prove

$$\sup_{f \in \overline {\mathcal C}}~ \|f\| ~<~ \infty?$$

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