Subspaces of \(\ell_2(X)\) and \(\text{Rad}(X)\) without local unconditional structure (Q2773406)

The following isomorphic characterization of Hilbert spaces is given:NEWLINENEWLINENEWLINEA Banach space \(X\) is isomorphic to a Hilbert space if and only if every subspace of \(\ell_2(X)\) and (or) of \(\text{Rad}(X)\) has a local unconditional structure.NEWLINENEWLINENEWLINEThe proof is based on the natural tensor product representations \(\ell^n_2(X)= \ell^n_2\otimes X\) and \(\text{Rad}(X)= \ell^n_2\otimes X\) and on a clever method to construct subspaces \(Z\) in the tensor products \(F\otimes \ell^n_2\otimes \ell^n_2\) (equipped with suitable norms) which have bad lust-constants under the assumption that the normalized basis \(\{f_1,\dots, f_n\}\) of \(F\) satisfies \(\|\sum_i f_i\|^2\geq 12n\) or \(\|\sum_i f_i\|^2\leq n/12\).