Coordinate systems and maximal order complexity in Banach lattices
Abstract
We resolve three questions arising from the study of coordinate systems and order convergence in Banach lattices. Firstly, every order (respectively, σ-order) basis (en) for a Banach lattice X=[en] is automatically a Schauder basis. In fact, the associated partial sum projections satisfy a maximal inequality, and hence every such basis is uniform. Secondly, we show that there is a separable Banach lattice X in which the set X↓0={(xn)∈ Xℕ:x1geqslant x2geqslant…geqslant 0=infnxn} is Π11-complete. Finally, we construct a separable Banach lattice Y for which the set Y↑∞ of increasing positive sequences which are not order bounded is Π11-complete. It follows that neither X↓0 nor Y↑∞ can be described in any simpler way than by the nonexistence of positive lower and upper bounds, respectively.
Related
- On the transformation of unbounded convergent sequences by linear operators on Banach lattices
- Norm attainment for multilinear operators and polynomials on Banach Spaces and Banach lattices
- DW-L-Weakly Compact Operators Between Banach Lattices
- Order continuity of induced homomorphisms between free Banach lattices
- Polish spaces of separable Banach lattices
BibTeX
@misc{aviles2026coordinate,
title = {{Coordinate systems and maximal order complexity in Banach lattices}},
author = {A. Avilés and C. Rosendal and M. A. Taylor and P. Tradacete},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33462},
eprint = {2610.33462},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33462},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Coordinate systems and maximal order complexity in Banach lattices
AU - A. Avilés
AU - C. Rosendal
AU - M. A. Taylor
AU - P. Tradacete
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33462
UR - https://arxiv.org/abs/2610.33462
AB - We resolve three questions arising from the study of coordinate systems and order convergence in Banach lattices. Firstly, every order (respectively, σ-order) basis (e_n) for a Banach lattice X=[e_n] is automatically a Schauder basis. In fact, the associated partial sum projections satisfy a maximal inequality, and hence every such basis is uniform. Secondly, we show that there is a separable Banach lattice X in which the set X_↓0={(x_n)∈ X^ℕ:x_1geqslant x_2geqslant…geqslant 0=inf_nx_n} is Π^1_1-complete. Finally, we construct a separable Banach lattice Y for which the set Y^↑∞ of increasing positive sequences which are not order bounded is Π^1_1-complete. It follows that neither X_↓0 nor Y^↑∞ can be described in any simpler way than by the nonexistence of positive lower and upper bounds, respectively.
ER -