EMBEDDABILITY OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES IS FINITELY DETERMINED

摘要

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 Xu 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.