Embeddability of locally finite metric spaces into Banach spaces is finitely determined

TL;DR

This paper proves that locally finite metric spaces that can be embedded into Banach spaces via finite subsets can also be embedded globally, extending previous results and using ultraproduct techniques.

Contribution

It establishes that finite embeddability into Banach spaces implies global embeddability for locally finite metric spaces, generalizing prior work.

Findings

Finite subsets' embeddings imply global embeddings.

Ultraproducts facilitate embedding extensions.

Results apply to bilipschitz and coarse embeddings.

Abstract

The main purpose of the paper is to prove the following results: Let $A$ be a locally finite metric space whose finite subsets admit uniformly bilipschitz embeddings into a Banach space $X$. Then $A$ admits a bilipschitz embedding into $X$. Let $A$ be a locally finite metric space whose finite subsets admit uniformly coarse embeddings into a Banach space $X$. Then $A$ admits a coarse embedding into $X$. These results generalize previously known results of the same type due to Brown-Guentner (2005), Baudier (2007), Baudier-Lancien (2008), and the author (2006, 2009). One of the main steps in the proof is: each locally finite subset of an ultraproduct $X^\mathcal{U}$ admits a bilipschitz embedding into $X$. We explain how this result can be used to prove analogues of the main results for other classes of embeddings.