, Volume 89, Issue 5, pp 419-429,
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Metrical characterization of super-reflexivity and linear type of Banach spaces

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 p norms.

Received: 2 August 2006, Revised: 10 April 2007