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

TL;DR

This paper revisits a lesser-known theorem of Ortyński, combining classical techniques to characterize subspaces of ll_p spaces with a given density, and applies these ideas to Lindenstrauss spaces for unique positioning.

Contribution

It introduces a novel combination of Ortyński's and Moreno-Plichko's methods to classify subspaces of ll_p spaces and extends these ideas to Lindenstrauss spaces for unique positioning.

Findings

Subspaces of ll_p with density al m have a specific form.

The techniques unify and extend existing classifications.

Unique positions for Lindenstrauss spaces are established.

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.