Unconditional basic sequences in spaces of large density

TL;DR

This paper investigates the existence of unconditional basic sequences in large-density Banach spaces, demonstrating that under certain set-theoretic assumptions, such sequences always exist in spaces of density _.

Contribution

It proves the relative consistency with GCH that every Banach space of density _ contains an unconditional basic sequence.

Findings

Unconditional basic sequences exist in Banach spaces of density   under certain set-theoretic assumptions.

The result is consistent with GCH for spaces of density  .

The study links set theory and Banach space theory in the context of large densities.

Abstract

We study the problem of the existence of unconditional basic sequences in Banach spaces of high density. We show, in particular, the relative consistency with GCH of the statement that every Banach space of density $\aleph_\omega$ contains an unconditional basic sequence.