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) One of the main steps in the proof is: each locally finite subset of an ultraproduct X 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.
The main object of the paper is to study the distance between Banach spaces introduced by Kadets. For Banach spaces X and Y , the Kadets distance is defined to be the infimum of the Hausdorff distance d(B X , B Y ) between the respective closed unit balls over all isometric linear embeddings of X and Y into a common Banach space Z. This is compared with the Gromov-Hausdorff distance which is defined to be the infimum of d(B X , B Y ) over all isometric embeddings into a common metric space Z. We prove continuity type results for the Kadets distance including a result that shows that this notion of distance has applications to the theory of complex interpolation.
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