This question comes to me when I read this paper : https://arxiv.org/pdf/1702.06049.pdf

Let ${\rm Lip}_0(\mathbb R^d)$ be the space of Lipschitz functions $F$ on $\mathbb R^d$ with $F(0)=0$. Then is ${\rm Lip}_0(\mathbb R^d)$ is a Banach space, with the norm defined as

$$\|F\|:=\sup_{x\neq y\in\mathbb R^d}~ \frac{|F(x)-F(y)|}{|x-y|}.$$

What is the dual space of ${\rm Lip}_0(\mathbb R^d)$?

PS: I am not familiar with the related literature, and my apologies if this question is not of research level. The motivation of this question is the following: Denote by $\Lambda\subset {\rm Lip}_0(\mathbb R^d)$ the subset consisting of functions $F$ of the form

$$F(x)=\sum_{i=1}^n f_i(v_i\cdot x),\quad \mbox{where } f_i\in {\rm Lip}_0(\mathbb R),~ v_i\in\mathbb R^d,~ n\ge 1.$$

I wish to show that $\Lambda$ is dense in ${\rm Lip}_0(\mathbb R^d)$. To prove that, I wish to argue by contradiction using the Hahn-Banach theorem, while I do not know the dual space of ${\rm Lip}_0(\mathbb R^d)$. Any answers or comments are highly appreciated.

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