Lipschitz-Free Spaces Over Ultrametric Spaces

Abstract

Abstract. We prove that the Lipschitz-free space over a separable ultrametric space has a monotone Schauder basis and is isomorphic to 1. This extends results of A. Dalet using an alternative approach.Mathematics Subject Classification. 46B03, 46B15, 54E35.

Lipschitz-free spaces over compact subsets of superreflexive spaces are weakly sequentially complete

“…Recall that for each k ∈ N the measure κ k lies in Ω, so it has a fixed representation (7) satisfying (8). For simplicity, we relabel the corresponding parameters as I k , a k i , p k i and q k i (1 i I k ).…”

“…Let us mention that other nontrivial examples of metric spaces whose Lipschitz‐free spaces are weakly sequentially complete, or even admit the Schur property, include uniformly discrete metric spaces, snowflaking of any metric space (both to be found in ), metric spaces that isometrically embed into an ‐tree , separable ultrametric spaces , countable proper metric spaces or metric spaces originating from ‐Banach spaces with a monotone finite‐dimensional decomposition .…”

Lipschitz-free Spaces on Finite Metric Spaces

“…Since it is well-known that ultrametrics can be isometrically embedded into weighted trees (see, for example, [12,Theorem 9], and also [21, Section 3]), we get also the following finite version of results of [15] and [13]:…”

“…We observe (Section 2) that the known fact (see [15], [13]) that Lipschitz free spaces on finite ultrametrics are close to ℓ 1 in the Banach-Mazur distance immediately follows from the result of Gupta [29] on Steiner points and the wellknown result on isometric embeddability of ultrametrics into weighted trees.…”

“…Proof of Theorem 4.2. Let us show, using Lemma 4.3, that in order to prove Theorem 4.2 it suffices to show that for each r ∈ N there exists n = n(r) ∈ N, C r ∈ Z(B n ), and a linear combination A r of vectors of the forms c(B 1 ) and their images under {E n } such that ||C r + A r || = 1 (13) and…”