Subspaces of $\ell _2(X)$ and ${\rm Rad}(X)$ without local unconditional structure

Ryszard A. Komorowski¹ ; Nicole Tomczak-Jaegermann²

¹Institute of Mathematics Wrocław Technical University 50-370 Wrocław, Poland
²Department of Mathematical Sciences University of Alberta Edmonton, Alberta Canada T6G 2G1

Studia Mathematica, Tome 149 (2002) no. 1, pp. 1-21

Résumé

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.

Détail

DOI : 10.4064/sm149-1-1

Keywords: shown banach space isomorphic hilbert space spaces ell mathop rad nolimits contain subspace without local unconditional structure therefore without unconditional basis moreover cotype infty subspace ell constructed without local unconditional structure dimensional unconditional decomposition hence basis