Orthogonality in nonseparable rearrangement-invariant spaces

Let $E$ be a nonseparable rearrangement-invariant space and let $E_0$ be the closure of the space of bounded functions in $E$. Elements of $E$ orthogonal to $E_0$, that is, elements $x\in E$, $x\ne 0$, such that $\|x\|_{E} \le\|x+y\|_{E}$ for each $y\in E_0$, are investigated. The set of orthogonal elements $\mathcal{O}(E)$ is characterized in the case when $E$ is a Marcinkiewicz or an Orlicz space. If an Orlicz space $L_M$ is considered with the Luxemburg norm, then the set $L_M\setminus (L_M)_0$ is the algebraic sum of $\mathcal{O}(L_M)$ and $(L_M)_0$. Each nonseparable rearrangement-invariant space $E$ such that $\mathcal{O}(E)\ne\varnothing$ is shown to contain an asymptotically isometric copy of the space $l_\infty$.
Bibliography: 17 titles.