Geometric Aspects of Functional Analysis pp 111-120 | Cite as

# A Hereditarily Indecomposable Space with an Asymptotic Unconditional Basis

## Abstract

A recent result of Maurey [3], extending the solution by Odell and Schlumprecht of the distortion problem [5], is that every uniformly convex Banach space with an unconditional basis has an arbitrarily distortable subspace. An important part of the proof, due to Milman and Tomczak-Jaegermann [4], is the statement that a space with a basis with no arbitrarily distortable subspace must have a subspace that is asymptotically *ℓ* _{ p }. This means that there is a constant *C* such that any normalized sequence of blocks *n* < *x* _{1} < *x* _{2} < ⋯ < *x* _{ n } is *C*-equivalent to the unit vector basis of *ℓ* _{ p } ^{ n } . (For the meaning of the symbol ‘<’ in this context, which is becoming standard, see the next section.) Loosely, such a space looks like *ℓ* _{ p } if one goes far enough along the basis. The argument is completed with the proof that if 1 < *p* < ∞ then an asymptotically *ℓ* _{ p } space with an unconditional basis is arbitrarily distortable.

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