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.



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?

  • 1
    $\begingroup$ Minor nitpick: it's Eells (two L's) $\endgroup$
    – Yemon Choi
    Commented Oct 1, 2021 at 15:56
  • 4
    $\begingroup$ 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. $\endgroup$ Commented Oct 1, 2021 at 17:09
  • 1
    $\begingroup$ 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. $\endgroup$ Commented Oct 2, 2021 at 1:59
  • 2
    $\begingroup$ 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. $\endgroup$ Commented Oct 5, 2021 at 16:25
  • 2
    $\begingroup$ 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 $\endgroup$ Commented Oct 7, 2021 at 18:26


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