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I started reading Nik Weaver's book on Lipschitz Algebras, where Arens-Eells spaces are identified as the pre-dual of the space of Lipschitz functions. The weak* convergence in $Lip_0(X)$ is then identified with point-wise convergence. The book also gives specific examples of Arens-Eels spaces, for example that $Æ(\mathbb{R})$ is isometric with $L^1(\mathbb{R})$, which is a separable space.

My question is: what can one say about the separability of $Æ(X)$, for example, for "simple" $X$, such as compact metric spaces (or even closed and bounded subspaces of $\mathbb{R}^n$)? The reason for asking is the following observation. On the one hand, we have the weak* compactness of the unit ball in $Lip_0(X)$ owing to Banach-Alaoglu. On the other hand, we have compactness (hence sequential compactness) of the unit ball of $Lip_0(X)$ as a subset of $C(X)$ owing to Arzela-Ascoli, and consequently the weak* sequential compactness of the unit ball of $Lip_0(X)$, since weak* convergence is equivalent to point-wise convergence.

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  • $\begingroup$ In general, if the dual of a Banach space is separable, so is the Banach space itself. Hence a sufficient criterion is that $\mathrm{Lip}_0(X)$ is separable. $\endgroup$
    – MaoWao
    Commented Dec 30, 2020 at 12:15

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If $X$ is separable then so is $A\!E(X)$.

Let $x_n$ be a sequence in $X$ with $x_n\to x$. Then $$\|x_n-x\|≤ d(x_n,x)$$ and $x_n -x \to 0$ in $A\!E(X)$. So if $X$ is separable with $\{y_n\mid n\in\Bbb N\}$ and let

$$m=\sum_i^n \lambda_i \ x_i$$ be a molecule. Suppose that $y_{n_i}$ is arbitrarily close to $x_i$ then $$\left\|m - \sum_i^n \lambda_i \ y_{n_i}\right\| ≤ \sum_i^n |\lambda_i| \,\left\|x_i -y_{n_i}\right\| $$ and note that the molecules with coefficients in $\{y_n\mid n\in\Bbb N\}$ are dense in the space of molecules hence in all of $A\!E(X)$. It follows that the rational combinations of $\{y_n\mid n\in\Bbb N\}$ are countable and dense in $A\!E (X)$.

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