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Products of Lipschitz-free spaces and applications

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Kaufmann P.

Studia Mathematica

2015

tom 226

nr 3

213-227

We show that, given a Banach space X, the Lipschitz-free space over X, denoted by ℱ(X), is isomorphic to $(∑_{n=1}^{∞}ℱ(X))_{ℓ₁}$. Some applications are presented, including a nonlinear version of Pełczyński's decomposition method for Lipschitz-free spaces and the identification up to isomorphism between ℱ(ℝⁿ) and the Lipschitz-free space over any compact metric space which is locally bi-Lipschitz embeddable into ℝⁿ and which contains a subset that is Lipschitz equivalent to the unit ball of ℝⁿ. We also show that ℱ(M) is isomorphic to ℱ(c₀) for all separable metric spaces M which are absolute Lipschitz retracts and contain a subset which is Lipschitz equivalent to the unit ball of c₀. This class includes all C(K) spaces with K infinite compact metric (Dutrieux and Ferenczi (2006) already proved that ℱ(C(K)) is isomorphic to ℱ(c₀) for those K using a different method).

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1 Studia Mathematica

1 Kaufmann P.

1 2015

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Products of Lipschitz-free spaces and applications