On free bases of Banach spaces

TL;DR

This paper introduces the concept of free bases in Banach spaces, compares Mol-equivalence with other metrics, and explores how various dimensions are preserved or not under this equivalence.

Contribution

It defines Mol-equivalence for metric spaces based on free bases and analyzes its relation to covering and Assouad dimensions, revealing new distinctions.

Findings

Mol-equivalent spaces share the same cech-Lebesgue dimension

Isomorphic Lipschitz-free spaces do not imply Mol-equivalence

Mol-equivalence does not preserve the metric Assouad dimension

Abstract

We call a closed subset M of a Banach space X a free basis of X if it contains the null vector and every Lipschitz map from M to a Banach space Y, which preserves the null vectors can be uniquely extended to a bounded linear map from X to Y. We then say that two complete metric spaces M and N are Mol-equivalent if they admit bi-Lipschitz copies M' and N', respectively that are free bases of a common Banach space satisfying span M'=span N'. In this note, we compare Mol-equivalence with some other natural equivalences on the class of complete metric spaces. The main result states that Mol-equivalent spaces must have the same \v{C}ech-Lebesgue covering dimension. In combination with the work of Godard, this implies that two complete metric spaces with isomorphic Lipschitz-free spaces need not be Mol-equivalent. Also, there exist non-homeomorphic Mol-equivalent metric spaces, and, in contrast with the covering dimension, the metric Assouad dimension is not preserved by Mol-equivalence.