Archiv der Mathematik

, Volume 89, Issue 5, pp 419-429

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

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

  • Florent BaudierAffiliated withLaboratoire de Mathématiques, UMR 6623, Université de Franche-Comté Email author 


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