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Schreier families and $\mathcal {F}$-(almost) greedy bases
Published online by Cambridge University Press: 14 June 2023
Abstract
Let $\mathcal {F}$ be a hereditary collection of finite subsets of $\mathbb {N}$. In this paper, we introduce and characterize $\mathcal {F}$-(almost) greedy bases. Given such a family $\mathcal {F}$, a basis $(e_n)_n$ for a Banach space X is called $\mathcal {F}$-greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$, $m \in \mathbb {N}$, and $G_m(x)$, we have
Furthermore, we provide several examples of bases that are nontrivially $\mathcal {F}$-greedy. For a countable ordinal $\alpha $, we consider the case $\mathcal {F}=\mathcal {S}_{\alpha }$, where $\mathcal {S}_{\alpha }$ is the Schreier family of order $\alpha $. We show that for each $\alpha $, there is a basis that is $\mathcal {S}_{\alpha }$-greedy but is not $\mathcal {S}_{\alpha +1}$-greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $\alpha < \beta $,
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society
Footnotes
The second author acknowledges the summer funding from the Department of Mathematics at the University of Illinois Urbana–Champaign.