Coordinate systems in Banach spaces and lattices

TL;DR

This paper uses descriptive set theory to analyze bases in Banach spaces and lattices, establishing new connections and properties under determinacy assumptions.

Contribution

It proves that under analytic determinacy, all $\sigma$-order bases in Banach lattices are uniform and Schauder, and explores filter Schauder bases in Banach spaces.

Findings

$\sigma$-order bases are uniform and Schauder in Banach lattices under determinacy.

Existence of Banach spaces with filter Schauder bases but no Schauder basis.

Every filter Schauder basis with an analytic filter is also with a Borel filter.

Abstract

Using methods of descriptive set theory, in particular, the determinacy of infinite games of perfect information, we answer several questions from the literature regarding different notions of bases in Banach spaces and lattices. For the case of Banach lattices, our results follow from a general theorem stating that (under the assumption of analytic determinacy), every $\sigma$-order basis $(e_n)$ for a Banach lattice $X=[e_n]$ is a uniform basis, and every uniform basis is Schauder. Moreover, the notions of order and $\sigma$-order bases coincide when $X=[e_n].$ Regarding Banach spaces, we address two problems concerning filter Schauder bases for Banach spaces, i.e., in which the norm convergence of partial sums is replaced by norm convergence along some appropriate filter on $\mathbb N$. We first provide an example of a Banach space admitting such a filter Schauder basis, but no ordinary Schauder basis. Secondly, we show that every filter Schauder basis with respect to an analytic filter is also a filter Schauder basis with respect to a Borel filter.