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.

  • $\begingroup$ In dimension one, by Rademacher's theorem, we can identify a Lipschitz function $f$ with an a.e. bounded function $f'$, and the Lipschitz norm of $f$ is just the (essential) supremum norm of $f'$. Thus, the dual of $\operatorname{Lip}_0(\mathbb{R})$ is isometrically isomorphic to the dual of $L^\infty(\mathbb{R})$. In higher dimensions, the description is likely similar, in terms of partial derivatives, but less explicit due to relations between the partial derivatives. $\endgroup$ – Mateusz Kwaśnicki Nov 24 '19 at 8:10
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    $\begingroup$ Precisely, the dual of $\text{Lip}_0(R^n)$ is isometrically isomorphic to the quotient $L^\infty(\mathbb{R}^n,\mathbb{R}^n)^*$ over the w* closure in $L^\infty(\mathbb{R}^n,\mathbb{R}^n)^*$ of $S:=\{g\in L^1(\mathbb{R}^n,\mathbb{R}^n): \text{div}(g)=0\}\subset L^1(\mathbb{R}^n,\mathbb{R}^n)\subset L^\infty(\mathbb{R}^n,\mathbb{R}^n)^*$ $\endgroup$ – Pietro Majer Nov 24 '19 at 9:09
  • $\begingroup$ So all bounded linear forms on ${\rm Lip}_0(\mathbb R^d)$ are of the form $f\mapsto <T,\nabla f>$ for $T\in L^\infty(\mathbb{R^d},\mathbb{R}^d)^*$, the representation being not unique. $\endgroup$ – Pietro Majer Nov 24 '19 at 10:33
  • $\begingroup$ @PietroMajer Thank you very kindly for the answer. Could you please write your answer in detail or provide the related references? $\endgroup$ – Neymar Nov 24 '19 at 12:35
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    $\begingroup$ Thank you, you are welcome; as you like, you're not obliged. As I understand here, if you find an answer useful you could up-vote it or even accept it, because the green color that it takes makes people in better mood, and more inclined to answer to other questions :) $\endgroup$ – Pietro Majer Nov 24 '19 at 14:59

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