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arxiv: 2406.11223 · v3 · submitted 2024-06-17 · 🧮 math.FA · math.LO

Coordinate systems in Banach spaces and lattices

Antonio Avil\'es , Christian Rosendal , Mitchell A. Taylor , Pedro Tradacete This is my paper

Pith reviewed 2026-05-24 00:15 UTC · model grok-4.3

classification 🧮 math.FA math.LO
keywords Banach latticesSchauder basesorder basesfilter Schauder basesanalytic determinacydescriptive set theoryBanach spaces
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The pith

Under analytic determinacy every σ-order basis in a Banach lattice is a uniform Schauder basis.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies analytic determinacy to show that several notions of bases coincide in Banach lattices generated by a sequence. Every σ-order basis is a uniform basis and every uniform basis is a Schauder basis. Order and σ-order bases are identical in this setting. It also constructs a Banach space that admits a filter Schauder basis but no ordinary Schauder basis and proves that any filter Schauder basis for an analytic filter is one for a Borel filter.

Core claim

Under the assumption of analytic determinacy, every σ-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 σ-order bases coincide when X=[e_n]. Regarding Banach spaces, there exists a Banach space admitting a filter Schauder basis but no ordinary Schauder basis, and every filter Schauder basis with respect to an analytic filter is also one with respect to a Borel filter.

What carries the argument

Analytic determinacy of infinite games of perfect information, applied to establish that σ-order convergence implies uniform and Schauder convergence for bases in lattices.

If this is right

  • Order, σ-order, uniform, and Schauder bases coincide for sequences that generate Banach lattices.
  • Filter Schauder bases exist in spaces that lack ordinary Schauder bases.
  • Any analytic-filter basis can be replaced by a Borel-filter basis without changing the convergence property.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • In set-theoretic models where analytic determinacy fails, Banach lattices may exist in which these basis notions remain distinct.
  • The constructed space offers a setting for studying bases that converge only along filters rather than in the usual sequential sense.
  • Descriptive-set-theoretic techniques used here could be tested on other coordinate systems such as frames or unconditional bases in related spaces.

Load-bearing premise

The lattice basis equivalences require that analytic determinacy holds.

What would settle it

A concrete Banach lattice generated by a σ-order basis that is not a Schauder basis, under the assumption of analytic determinacy.

Figures

Figures reproduced from arXiv: 2406.11223 by Antonio Avil\'es, Christian Rosendal, Mitchell A. Taylor, Pedro Tradacete.

Figure 1
Figure 1. Figure 1: Implications between convergence types. Here the notation ∀∞n means for all but finitely many n, i.e., ∃N ∀n > N, while zm ↓ 0 and zµ ↓ 0 mean that (zm) and (zµ) are decreasing and have infimum 0. It can be shown that, in all cases above, the limit is unique whenever it exists. Thus, if C is one of the above notions of convergence and P∞ n=1 xn is a series in X, we can unambiguously write x = C X∞ n=1 xn t… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper applies descriptive set theory, particularly analytic determinacy, to resolve questions on bases in Banach spaces and lattices. Under analytic determinacy, it proves that every σ-order basis (e_n) for a Banach lattice X=[e_n] is a uniform basis, every uniform basis is Schauder, and order and σ-order bases coincide on such X. For general Banach spaces, it constructs an example admitting a filter Schauder basis but no ordinary Schauder basis, and shows every filter Schauder basis w.r.t. an analytic filter is also one w.r.t. a Borel filter.

Significance. If the results hold, they provide conditional resolutions to open questions in the literature by connecting Banach lattice and space theory with determinacy assumptions from set theory. The explicit construction separating filter and ordinary Schauder bases, together with the analytic-to-Borel reduction, offers concrete advances; the lattice equivalences are parameter-free once the determinacy hypothesis is fixed.

minor comments (3)
  1. [Introduction] Introduction, paragraph 2: the phrase 'several questions from the literature' is not followed by an explicit list of the targeted problems or citations; adding a short enumerated list would clarify the scope.
  2. [Section 3] The statement of the main lattice theorem (presumably Theorem 3.1 or equivalent) conditions all equivalences on analytic determinacy; a brief remark on whether any of the implications survive in ZFC alone would strengthen the presentation.
  3. [Section 4] The filter-basis example relies on a specific Banach space construction whose details appear in §4; a short diagram or explicit norm formula for the partial-sum operators would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; results rest on external assumptions and explicit constructions

full rationale

The paper derives its lattice equivalences conditionally on analytic determinacy (an external axiom about games) and its Banach space examples via explicit constructions plus definability properties of filters. No step reduces a claimed prediction or uniqueness result to a self-definition, fitted input, or self-citation chain; the central claims remain independent of the target notions by construction. This is the expected non-finding for a paper whose derivations are self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The lattice results rest on the external assumption of analytic determinacy; the filter results rest on the existence of a particular Banach-space construction whose details are not supplied in the abstract. No free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Analytic determinacy (every analytic game of perfect information is determined)
    Invoked to obtain the equivalences between σ-order, uniform, and Schauder bases in Banach lattices.

pith-pipeline@v0.9.0 · 5726 in / 1418 out tokens · 24320 ms · 2026-05-24T00:15:31.455227+00:00 · methodology

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Reference graph

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