Metrical characterization of super-reflexivity and linear type of Banach spaces
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- Baudier, F. Arch. Math. (2007) 89: 419. doi:10.1007/s00013-007-2108-4
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 dp induced by the ℓp norms.