**Context**

*Arens-Eells space.* Let $X$ be a separable pointed metric space with base point $e$. An *elementary molecule* is defined as follows (Nik Weaver, Lipschitz Algebras, 2nd ed.)
$$
m_{pq} := \delta_p - \delta_q,
$$

where $\delta_p, \delta_q$ are delta-functions placed at $p,q$. The Arens-Eells space $AE(X)$ (also known as the Lipschitz-free space) is the completion of the linear span of elementary molecules with respect to the Arens-Eells norm $$ \|{m}\|_{AE} := \inf \left\{\sum_{i=1}^n |{a_i}| d(p_i,q_i) \colon m = \sum_{i=1}^n a_i m_{p_iq_i} \right\}, $$

where $d(p,q)$ is the distance between $p,q \in X$.

The dual of the Arens-Eells space is the $Lip_0(X)$ space of all Lipschitz functions on $X$ vanishing at $e$ equipped with the following norm $$ \|f\|_{Lip_0} := Lip(f), $$ where $Lip(f)$ denotes the Lipschitz constant.

*Schauder bases.* Let $U$ be a Banach space. A countable system $\{u_i\}_{i\in\mathbb N} \subset U$ is called a basis (or Schauder basis) if for any $u \in U$ there exists a unique sequence of coefficient functionals $\{c_i(u)\}_{i\in\mathbb N} \subset \mathbb R$ such that
$$
u =\sum_{i=1}^\infty c_i(u) u_i
$$
and the sum converges strongly. The coefficient functionals can be represented with a system $\{v_i\}_{i\in\mathbb N} \subset U^*$ in the dual of $U$ and
$$
u =\sum_{i=1}^\infty \langle v_i,u\rangle u_i,
$$
where $\langle \cdot, \cdot \rangle $ denotes the duality pairing. It is easy to check that the system $\{u_i,v_i\}_{i\in\mathbb N}$ is *biorthogonal*, i.e.
$$
\langle v_j,u_i\rangle = \delta_{ij}
$$
holds, where $\delta_{ij}$ is the Kronecker delta.

NB: Not all separable Banach spaces have a basis (Enflo, 1973). However, most "common" spaces do.

**Question 1**

Does the Arens-Eells space over a compact metric space $X$ have a Schauder basis and if it does, how can it be expressed using elementary molecules?

**Thoughts**
One would think that elementary molecules
$$
m_{ij} := \delta_{p_i} - \delta_{p_j}
$$
supported on the countable dense system $\{p_i\}_{i\in\mathbb N} \subset X$ (which exists because $X$ is separable) would form a basis. Let $\{f_{ij}\}_{i,j\in\mathbb N} \subset Lip_0(X)$ be the corresponding coefficients functionals. The biorthogonality condition would then imply that
$$
\langle f_{ij},m_{kl}\rangle = f_{ij}(p_k) - f_{ij}(p_l) = 0 \quad \text{for $(i,j) \neq (k,l)$}
$$
and
$$
\langle f_{ij},m_{ij}\rangle = f_{ij}(p_i) - f_{ij}(p_j) = 1,
$$
which is a condition I struggle to interpret, epsecially since $(p_k,p_l)$ can be arbitrary close to $(p_i,p_j)$.

Any help will be much appreciated.

$\quad$

**Update**

Having read the Godefroy&Ozawa paper suggested in the comments below, I realised that the situation is subtler than I thought. In the generality I initially asked the question, the answer is "no" -- the space may fail the Approximation Propoerty and hence not have a basis. However, in special cases, the Approximation Property (and even the Bounded Approximation Property) is guaranteed -- for example, if $X$ is a doubling metric space (Lancien&Pernecka). This does not imply the existence of a basis, however (although Borel-Mathurin shows existence of a Schauder decomposition in $AE(\mathbb R^n)$ -- a related, but as far as I understand weaker property).

Let me now ask a more specific question.

**Question 2**

Let $X \subset \mathbb R^n$ be a compact set (obviously, a doubling metric space). Does $AE(X)$ have a basis and if yes, how is it expressed using elementary molecules?

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