A Coarse-Lipschitz Embedding of c₀ into a Separable Dual Banach Space

2026/08/04 by Bunyamin Sari
Mathematics · #math.FA

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arxiv created 2026/08/04 · arxiv updated 2026/08/06

Abstract

Let G=ℤ^<ω⊂ c₀ \qquadand G_R=G∩ R B_c0, R∈ℕ, where the metric d is inherited from c₀. On each G_R we construct a commuting family of retractions onto finite initial segments of a special ordering of G_R, with Lipschitz constant at most two. This yields a boundedly complete basis of the Lipschitz-free space F(G_R) whose basis constant is at most two, and 2R-equivalent to the unit vector basis of ℓ₁. Thus the space is 2-isomorphic to a dual space. The constant two is sharp: the radius-two grid G₂ does not embed with distortion strictly less than two into a separable dual Banach space. Using Kalton's annular decomposition, we then embed F(G) into a fixed separable dual space with distortion at most 2(1+ε) for every ε>0. Since G is an integer net in c₀, this gives a coarse-Lipschitz embedding of c₀ into a separable dual. The optimal coarse-Lipschitz distortion, understood as an infimum over all separable dual targets, is equal to two.

Citations

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