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arXiv · 2405.03556

On free bases of Banach spaces

E. Pernecká·J. Spěvák
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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.

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E. Pernecká, J. Spěvák. 2024-05-06. On free bases of Banach spaces. https://arxiv.org/abs/2405.03556

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