# Schauder basis in the Arens-Eells space

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.

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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?

• Minor nitpick: it's Eells (two L's) Commented Oct 1, 2021 at 15:56
• The paper "Free Banach spaces and the approximation properties" by G. Godefroy and N. Ozawa (Proc. Am. Math. Soc. 142, No. 5, 1681-1687 (2014)) gives an example of a Lipschitz free space over a compact metric space without the approximation property, in particular this space fails to have a Schauder basis. See Zbl 1291.46013. Commented Oct 1, 2021 at 17:09
• There is an arxiv version of the Godefroy-Ozawa paper. The required classical fact that every separable Banach space is the closure of the span of a compact convex set has been asked about before here. Commented Oct 2, 2021 at 1:59
• On page 131 of his book, Heil says, In Theorem 4.13 we will prove the nontrivial fact that every basis for a Banach space is a Schauder basis.'' It is not clear to me why he makes the distinction basis / Schauder basis to begin with. Commented Oct 5, 2021 at 16:25
• Comment 2: Relevant papers for Question 2: Pernecká, Eva; Smith, Richard J., The metric approximation property and Lipschitz-free spaces over subsets of $\mathbb R^N$; J. Approx. Theory 199, 29-44 (2015). Zbl 1333.46017, Hájek, Petr; Pernecká, Eva, On Schauder bases in Lipschitz-free spaces. J. Math. Anal. Appl. 416, No. 2, 629-646 (2014). Zbl 1319.46010 and other papers by Eva Pernecka Commented Oct 7, 2021 at 18:26