An ordering for the Banach spaces (Q789669)

An ''ordering'' on the class of all Banach spaces is defined as follows: If X and Y are Banach spaces, then we say \(X\prec Y\) if \(X=\cap T^{**- 1}[Y],\) where the intersection is over all bounded linear operators T:\(X\to Y\). This holds if there is a Tauberian operator from X to Y, but also in other cases. Certain special cases are characterized: \(X\prec c_ 0\) if and only if X has the property of Mazur \((weak^*\) continuous functionals on \(X^*\) are in X). \(X\prec \ell_{\infty}\) if and only if any \(x^{**}\in X^{**}\) that is \(weak^*\) continuous on bounded \(weak^*\) separable subsets of \(X^*\) is in X. \(\ell_ 1\prec X\) if and only if X is not reflexive. Connections are discussed with the theory of the Pettis integral, and with uniqueness of preduals.