On the hereditary proximity to 1

In the first part of the paper we present and discuss concepts of local and asymptotic hereditary proximity to 1. The second part is devoted to a complete separation of the hereditary local proximity to 1 from the asymptotic one. More precisely for every countable ordinal ξ we construct a separable Hereditarily Indecomposable reflexive space Xξ such that every infinite-dimensional subspace of it has Bourgain 1- index greater than ωξ and the space itself has no 1-spreading model. © 2011 Elsevier Inc. All rights reserved