Related papers: Products of Lipschitz-free spaces and applications

We consider subsets $S$ of a metric space $M$ such that Lipschitz mappings defined on $S$ can be extended to Lipschitz mappings on $M$, and we show that the union of such subsets has the same property under appropriate geometric conditions.…

We give several structural results concerning the Lipschitz-free spaces $\mathcal F(M)$, where $M$ is a metric space. We show that $\mathcal F(M)$ contains a complemented copy of $\ell_1(\Gamma)$, where $\Gamma=\text{dens}(M)$. If $\mathcal…

We present a way to turn an arbitrary (unbounded) metric space $\mathcal{M}$ into a bounded metric space $\mathcal{B}$ in such a way that the corresponding Lipschitz-free spaces $\mathcal{F}(\mathcal{M})$ and $\mathcal{F}(\mathcal{B})$ are…

We prove that the Lipschitz-free space over a Banach space $X$ of density $\kappa$, denoted by $\mathcal{F}(X)$, is linearly isomorphic to its $\ell_1$-sum $\left(\bigoplus_{\kappa}\mathcal{F}(X)\right)_{\ell_1}$. This provides an extension…

We find general conditions under which Lipschitz-free spaces over metric spaces are isomorphic to their infinite direct $\ell_1$-sum and exhibit several applications. As examples of such applications we have that Lipschitz-free spaces over…

We set up a descriptive set-theoretic framework to study Lipschitz-free spaces and use the reduction argument of Bossard to prove several results. We prove two universality results: if a separable Banach space is isomorphically universal…

In this note we study the structure of Lipschitz-free Banach spaces. We show that every Lipschitz-free Banach space over an infinite metric space contains a complemented copy of $\ell_1$. This result has many consequences for the structure…

We prove that, for each Banach space $X$ which is isomorphic to its hyperplanes, the Lipschitz-free spaces over $X$ and over its sphere are isomorphic.

We prove the result stated in the title. This provides a first example of an infinite-dimensional Banach space whose Lipschitz free space is isomorphic to the free space of a compact set.

In the present paper we introduce and study the Lipschitz retractional structure of metric spaces. This topic was motivated by the analogous projectional structure of Banach spaces, a topic that has been thoroughly investigated. The more…

We study several properties and applications of the ultrapower $M_{\mathcal U}$ of a metric space $M$. We prove that the Lipschitz-free space $\mathcal F(M_{\mathcal U})$ is finitely representable in $\mathcal F(M)$. We also characterize…

The goal of this paper is to study geometric and extremal properties of the convex body $B_{\mathcal F(M)}$, which is the unit ball of the Lipschitz-free Banach space associated with a finite metric space $M$. We investigate $\ell_1$ and…

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in…

We prove that for any separable Banach space $X$, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to $X$. As a consequence we give an…

We prove that the Lipschitz-free space over a separable ultrametric space has a monotone Schauder basis and is isomorphic to $\ell_1$. This extends results of A. Dalet using an alternative approach.

We prove that the Lipschitz-free space over a countable compact metric space is isometric to a dual space and has the metric approximation property.

In this paper we study $\ell_1$-like properties for some Lipschitz-free spaces. The main result states that, under some natural conditions, the Lipschitz-free space over a proper metric space linearly embeds into an $\ell_1$-sum of finite…

The isometric universality of the spaces $C(K)$ for $K$ a non scattered Hausdorff compact does not take into account the ``quality'' of the representation. Indeed, the existence of an isometric copy of a separable Banach space $X$ into…

In our note we show the very close connection between the existence of a Finite Dimensional Decomposition (FDD for short) for a separable Banach space $X$ and the existence of a Lipschitz retraction of $X$ onto a small (in a certain precise…

We show that, for a separable and complete metric space $M$, the Lipschitz-free space $\mathcal F(M)$ embeds linearly and almost-isometrically into $\ell_1$ if and only if $M$ is a subset of an $\mathbb R$-tree with length measure 0.…