For a Banach space X with an unconditional basic sequence, one of the following regular-irregular alternatives holds: either X contains a subspace isomorphic to ℓ₂, or X contains a subspace which has an unconditional finite-dimensional decomposition, but does not admit such a decomposition with a uniform bound for the dimensions of the decomposition. This result can be viewed in the context of Gowers' dichotomy theorem.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-1-4,
author = {Razvan Anisca},
title = {On the structure of Banach spaces with an unconditional basic sequence},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {67-85},
zbl = {1136.46009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-1-4}
}
Razvan Anisca. On the structure of Banach spaces with an unconditional basic sequence. Studia Mathematica, Tome 178 (2007) pp. 67-85. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-1-4/