On the number of permutatively inequivalent basic sequences in a Banach space

Let X be a Banach space with a Schauder basis (en)n∈N. The relation E0 is Borel reducible to permutative equivalence between normalized block-sequences of (en)n∈N or X is c0 or p saturated for some 1 p <+∞. If (en)n∈N is shrinking unconditional then either it is equivalent to the canonical basis of c0 or p, 1 < p <+∞, or the relation E0 is Borel reducible to permutative equivalence between sequences of normalized disjoint blocks of X or of X∗. If (en)n∈N is unconditional, then either X is isomorphic to 2, or X contains 2ω subspaces or 2ω quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases. © 2006 Elsevier Inc. All rights reserved.