A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if E₀ does not reduce to isomorphism of the subspaces of a space, in particular, if the subspaces of the space admit a classification up to isomorphism by real numbers, then any subspace with an unconditional basis is isomorphic to its square and hyperplanes, and the unconditional basis has an isomorphically homogeneous subsequence.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm183-3-5,
author = {Christian Rosendal},
title = {Incomparable, non-isomorphic and minimal Banach spaces},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {253-274},
zbl = {1078.46006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm183-3-5}
}
Christian Rosendal. Incomparable, non-isomorphic and minimal Banach spaces. Fundamenta Mathematicae, Tome 184 (2004) pp. 253-274. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm183-3-5/