Abstract.
We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain’s result who gave a metrical characterization of super-reflexivity in Banach spaces in terms of uniform embeddings of the finite trees. A characterization of the linear type for Banach spaces is given using the embedding of the infinite tree equipped with the metrics d p induced by the ℓ p norms.
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Received: 2 August 2006, Revised: 10 April 2007
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Baudier, F. Metrical characterization of super-reflexivity and linear type of Banach spaces. Arch. Math. 89, 419–429 (2007). https://doi.org/10.1007/s00013-007-2108-4
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DOI: https://doi.org/10.1007/s00013-007-2108-4