On a Property of Rearrangement Invariant Spaces whose Second Köthe Dual is Nonseparable

We study the family of rearrangement invariant spaces $E$ containing subspaces on which the $E$-norm is equivalent to the $L_1$-norm and a certain geometric characteristic related to the Kadec–Pełcziński alternative is extremal. We prove that, after passing to an equivalent norm, any space with nonseparable second Köthe dual belongs to this family. In the course of the proof, we show that every nonseparable rearrangement invariant space $E$ can be equipped with an equivalent norm with respect to which $E$ contains a nonzero function orthogonal to the separable part of $E$.