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

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Y</mml:mi> </mml:math> be two infinite-dimensional Banach spaces. If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is crudely finitely representable in every finite-codimensional subspace of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Y</mml:mi> </mml:math> , then any proper subset of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> almost bi-Lipschitz embeds into <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Y</mml:mi> </mml:math> , in a sense quite close to that of F. Baudier and G. Lancien (see [1] and [2]). This is an extension of a result proved by M.I. Ostrovskii for locally finite subsets [9].