# Arens-Eells spaces

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.

• 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. Commented Dec 30, 2020 at 12:15

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 $$A\!E (X)$$.