Subspaces of $\ell _2(X)$ and ${\rm Rad}(X)$ without local unconditional structure
Volume 149 / 2002
Studia Mathematica 149 (2002), 1-21
MSC: 46B03, 46B07.
DOI: 10.4064/sm149-1-1
Abstract
It is shown that if a Banach space $X$ is not isomorphic to a Hilbert space then the spaces $\ell _2(X)$ and $\mathop {\rm Rad}\nolimits (X)$ contain a subspace $Z$ without local unconditional structure, and therefore without an unconditional basis. Moreover, if $X$ is of cotype $r <\infty $, then a subspace $Z$ of $\ell _2(X)$ can be constructed without local unconditional structure but with 2-dimensional unconditional decomposition, hence also with basis.
