Dvoretzky-Type Theorem for Locally Finite Subsets of a Hilbert Space

Abstract

The main result of the paper: Given any epsilon > 0, every locally finite subset of l(2) admits a (1 + epsilon)-bilipschitz embedding into an arbitrary infinite-dimensional Banach space. The result is based on two results which are of independent interest: (1) A direct sum of two finite-dimensional Euclidean spaces contains a sub-sum of a controlled dimension which is epsilon-close to a direct sum with respect to a 1-unconditional basis in a two-dimensional space. (2) For any finite-dimensional Banach space Y and its direct sum X with itself with respect to a 1-unconditional basis in a two-dimensional space, there exists a (1 + epsilon)-bilipschitz embedding of Y into X which on a small ball coincides with the identity map onto the first summand and on the complement of a large ball coincides with the identity map onto the second summand.