Let $C(\mathbb R^d)$ be the space of continuous functions on $\mathbb R^d$, and $C_{lip}(\mathbb R^d)\subset C(\mathbb R^d)$ be the subspace of Lipschitz functions. We endow $C_{lip}(\mathbb R^d)$ with the following topology: $(f_n)_{n\ge 1} \subset C_{lip}(\mathbb R^d)$ converges to $f\in C_{lip}(\mathbb R^d)$ iff

$$\lim_{n\to\infty} \left\{\left|\int_{\mathbb R^d}(f_n-f)(x)u(x)dx\right| + \left|\int_{\mathbb R^d}\nabla(f_n-f)(x)\cdot w(x)dx\right|\right\} = 0,$$

for all $u:\mathbb R^d\to\mathbb R$ and $w:\mathbb R^d\to\mathbb R^d$ satisfying

$$\int_{\mathbb R^d}|u(x)|(1+|x|)dx<\infty \quad\mbox{and}\quad \int_{\mathbb R^d}|w(x)|dx<\infty.$$

Is this topology metrizable?

PS: The motivation is to apply Baire's theorem. Any answers or comments are highly appreciated.

PS2: Thank Iosif Pinelis for the answer. I wish to ask a further question: Consider the completion $\overline{C}_{lip}(\mathbb R^d)$ of $C_{lip}(\mathbb R^d)$ w.r.t. this metric. Could we show that any **linear continuous** function $T: \overline{C}_{lip}(\mathbb R^d)\to\mathbb R$ must be of the form

$$T(f)=\int_{\mathbb R^d}f(x)u(x)dx+\int_{\mathbb R^d}\nabla f(x)w(x)dx?$$