A Hereditarily Indecomposable Space with an Asymptotic Unconditional Basis
A recent result of Maurey , extending the solution by Odell and Schlumprecht of the distortion problem , 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 , 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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