Almost bi-Lipschitz embeddings and proper subsets of a Banach space -- An extension of a theorem by M.I. Ostrovskii

TL;DR

This paper extends Ostrovskii's theorem by showing that under certain conditions, proper subsets of an infinite-dimensional Banach space can almost bi-Lipschitz embed into another Banach space, broadening the scope of embedding results.

Contribution

It generalizes Ostrovskii's result from locally finite subsets to proper subsets, under the condition of crude finite representability in finite-codimensional subspaces.

Findings

Proper subsets of X embed into Y with almost bi-Lipschitz maps.

Extension of Ostrovskii's theorem to broader classes of subsets.

Conditions involving finite representability are key for embeddings.

Abstract

Let X and Y be two infinite-dimensional Banach spaces. If X is crudely finitely representable in every finite-codimensional subspace of Y, then any proper subset of X almost bi-Lipschitz embeds into Y, in a sense quite close to that of F. Baudier and G. Lancien. This is an extension of a result proved by M.I. Ostrovskii for locally finite subsets.