The distortion problem

1994/01/01 by Edward Odell, Thomas Schlumprecht · 22 citations
Mathematics · #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems #Advanced Banach Space Theory

paper · pdf · doi:10.1007/bf02398436

Abstract

Abstract. We prove that Hilbert space is distortable and, in fact, arbitrarily distortable. This means that for all λ> 1 there exists an equivalent norm | · | on ℓ2 such that for all infinite dimensional subspaces Y of ℓ2 there exist x, y ∈ Y with ‖x‖2 = ‖y‖2 = 1 yet |x |> λ|y|. We also prove that if X is any infinite dimensional Banach space with an unconditional basis then the unit sphere of X and the unit sphere of ℓ1 are uniformly homeomorphic if and only if X does not contain ℓ n ∞’s uniformly. 1.

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