A forgotten theorem of Orty\'nski, not far off

Abstract

We combine techniques of Orty\'nski \cite{orty} and Moreno and Plichko \cite{moreplic} to show that, for $0<p<\infty$, every subspace of $\ell_p(\Gamma)$ having density character $\mathfrak m$ has the form $\ell_p(\mathfrak m, H_m)$ for some family of subspaces $H_m\subset \ell_p$. The ideas of Orty\'nski are then applied to obtain unique positions for Lindenstrauss spaces.