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arXiv · 2508.00490

On Banach envelopes and duals of Lipschitz-free $p$-spaces for $0<p<1$

Fernando Albiac·Jose L. Ansorena
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Abstract

With the aim to better understand the intricate geometry of the class of Lipschitz free $p$-spaces $\mathcal{F}_p(\mathcal{M})$ when $0<p<1$, in this note we study their Banach envelopes and prove that if $0<p<1$ and $ \mathcal{M}$ is a metric space then the Banach envelope map of $\mathcal{F}_p(\mathcal{M})$ is one-to-one, thus solving in the positive a problem raised by Kalton in [F. Albiac and N. J. Kalton, Lipschitz structure of quasi-Banach spaces, Israel J. Math. 170 (2009), 317-335]. This property has important applications to the linear structure of this family of spaces, being the most immediate one that the dual space of $ \mathcal{F}_p(\mathcal{M})$ separates the points of $\mathcal{F}_p(\mathcal{M})$.

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Fernando Albiac, Jose L. Ansorena. 2025-08-01. On Banach envelopes and duals of Lipschitz-free $p$-spaces for $0<p<1$. https://arxiv.org/abs/2508.00490

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