Basic and quasibasic subspaces in dual Banach spaces (Q756648)

Let X be a Banach space and \(X^*\) be its dual. A subspace \(M\subset X^*\) is said to be quasi-basic if it contains all biorthogonal functionals of some (linear) operational basis of X. A subspace \(M\subset X^*\) is said to be basic if it contains all biorthogonal functionals of some basis of X. The existence of spaces with a basis the duals of which contain quasi- basic subspaces which are not basic is proved. The examples of such spaces are \(\ell_ 1\), \(L_ 1(0,1)\).