On Unconditionality and Higher-Order Schreier Unconditionality

Abstract

Let $X$ be a Banach space, $(e_n{)}_{n=1}^{\infty }$ be its basis, and $S_{\hat{I}}\pm$ be a Schreier family of order alpha. We introduce Condition A which is a weaker version of the Continuum Hypothesis. Granted Condition A, we show that if the basis $(e_n)$ is $S_{\hat{I}}\pm$-unconditional for every countable ordinal alpha, then it is unconditional.