Archiv der Mathematik

, Volume 89, Issue 5, pp 419–429 | Cite as

Metrical characterization of super-reflexivity and linear type of Banach spaces

Open Access


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.

Mathematics Subject Classification (2000).

46B20 51F99 


Super-reflexivity trees linear type metric embedding 

Copyright information

©  2007

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques, UMR 6623Université de Franche-ComtéBesançon, cedexFrance

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