Orthogonal elements in nonseparable rearrangement invariant spaces

Let $E$ be a nonseparable rearrangement invariant space, and let $E_0$ denote the closure of the set of all bounded functions in $E$. We study elements of $E$ orthogonal to the subspace $E_0$, i.e., elements $x\in E$ such that $\|x\|_E\le\|x+y\|_E$ for any $y\in E_0$.