On Unconditionality and Higher-Order Schreier Unconditionality

TL;DR

The paper addresses whether $S_\alpha$-unconditionality for every countable ordinal $\alpha$ implies unconditionality, introducing Condition A as a weaker-than-CH hypothesis. It develops an ordinal-tree framework via the function $F(n,\alpha)$ to connect growth patterns of Schreier sets to unconditionality, and proves that under Condition A, $S_\alpha$-unconditionality for all $\alpha$ indeed yields unconditionality. It then analyzes the necessity of Condition A by constructing a framework with $S_{\omega_1}$ and showing how the failure of Condition A would influence the landscape of possible counterexamples, thereby clarifying the boundaries of the main result. Overall, the work provides a conditional positive resolution to an antecedent question about higher-order Schreier unconditionality and highlights the role of set-theoretic assumptions in Banach space structure theory.

Abstract

Let $X$ be a Banach space, $(e_n)_{n=1}^\infty$ be its basis, and $S_α$ 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 $S_α$-unconditional for every countable ordinal alpha, then it is unconditional.

Theorems & Definitions (36)

• Definition 1.1
• Definition 1.2
• Lemma 1.3
• Definition 1.5
• Lemma 1.6
• Lemma 1.7
• Lemma 2.1
• proof
• Corollary 2.2
• Lemma 2.3