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ZOO OF IDEAL SCHAUDER BASES

Published online by Cambridge University Press:  05 September 2025

ADAM KWELA*
Affiliation:
INSTITUTE OF MATHEMATICS FACULTY OF MATHEMATICS PHYSICS AND INFORMATICS UNIVERSITY OF GDAŃSK UL. WITA STWOSZA 57 80-308 GDAŃSK POLAND URL: https://mat.ug.edu.pl/~akwela
JAROSŁAW SWACZYNA
Affiliation:
INSTITUTE OF MATHEMATICS ŁÓDŹ UNIVERSITY OF TECHNOLOGY ALEJE POLITECHNIKI 8 93-590 ŁÓDŹ POLAND E-mail: jaroslaw.swaczyna@p.lodz.pl

Abstract

We investigate the notion of ideal (equivalently: filter) Schauder basis of a Banach space. We do so by providing bunch of new examples of such bases that are not the standard ones, especially within classical Banach spaces ($\ell _p$, $c_0$, and James’ space). Those examples lead to distinguishing and characterizing ideals (equivalently: filters) in terms of Schauder bases. We investigate the relationship between possibly basic sequences and ideals (equivalently: filters) on the set of natural numbers.

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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

REFERENCES

Albiac, F. and Kalton, N. J., Topics in Banach Space Theory, second ed., With a foreword by Gilles Godefory, Graduate Texts in Mathematics, 233, Springer, Cham, 2016.10.1007/978-3-319-31557-7CrossRefGoogle Scholar
Avilés, A., Rosendal, C., Taylor, M. A., and Tradacete, P., Coordinate systems in Banach spaces and lattices, preprint. avaible at https://arxiv.org/abs/2406.11223.Google Scholar
Borodulin-Nadzieja, P. and Farkas, B., Analytic P-ideals and Banach spaces . Journal of Functional Analysis, vol. 279 (2020), no. 8, p. 108702.10.1016/j.jfa.2020.108702CrossRefGoogle Scholar
Connor, J., Ganichev, M., and Kadets, V., A characterization of Banach spaces with separable duals via weak statistical convergence . Journal of Mathematical Analysis and Applications, vol. 244 (2000), pp. 251261.10.1006/jmaa.2000.6725CrossRefGoogle Scholar
Debs, G. and Saint Raymond, J., Filter descriptive classes of Borel functions . Fundamenta Mathematicae, vol. 204 (2009), pp. 189213.10.4064/fm204-3-1CrossRefGoogle Scholar
de Rancourt, N., Kania, T., and Swaczyna, J., Continuity of coordinate functionals of filter bases in Banach spaces . Journal of Functional Analysis, vol. 284 (2023), no. 9, p. 109869.CrossRefGoogle Scholar
Farah, I., Analytic quotients. theory of lifting for quotients over analytic ideals on integers . Memoirs of the American Mathematical Society, vol. 148 (2000), p. 702.10.1090/memo/0702CrossRefGoogle Scholar
Fast, H., Sur la convergence statistique . Colloquium Mathematicum, vol. 2 (1951), pp. 4144.10.4064/cm-2-3-4-241-244CrossRefGoogle Scholar
Fridy, J. A., On statistical convergence . Analysis, vol. 5 (1985), pp. 301313.10.1524/anly.1985.5.4.301CrossRefGoogle Scholar
Ganichev, M. and Kadets, V., Filter convergence in banach spaces and generalized bases , General Topology in Banach Spaces (Banakh, T., editors), Nova Science Publishers, Inc., Huntington NY, 2001, pp. 6169.Google Scholar
Guzmán-González, O. and Meza-Alcántara, D., Some structural aspects of the Katětov order on Borel ideals . Order, vol. 33 (2016), pp. 189194.CrossRefGoogle Scholar
Kania, T. and Swaczyna, J., Large cardinals and continuity of coordinate functionals of filter bases in Banach spaces . Bulletin of the London Mathematical Society, vol. 53 (2021), no. 1, pp. 231239.CrossRefGoogle Scholar
Kochanek, T., Bases with brackets and with individual brackets in Banach spaces . Studia Mathematica, vol. 211 (2012), pp. 259268.10.4064/sm211-3-7CrossRefGoogle Scholar
Kostyrko, P., Šalat, T., and Wilczyński, W., I-convergence . Real Analysis Exchange, vol. 26 (2000), pp. 669685.10.2307/44154069CrossRefGoogle Scholar
Mazur, K., ${F}_{\sigma }$ -ideals and ${\omega}_1{\omega}_1^{\ast }$ -gaps in the Boolean algebras $P\left(\omega \right)/I$ . Fundamenta Mathematicae, vol. 138 (1991), pp. 103111.10.4064/fm-138-2-103-111CrossRefGoogle Scholar
Šalát, T., On statistically convergent sequences of real numbers . Mathematica Slovaca, vol. 30 (1980), pp. 139150.Google Scholar
Solecki, S., Analytic ideals . The Bulletin of Symbolic Logic, vol. 2 (1996), pp. 339348.10.2307/420994CrossRefGoogle Scholar
Solecki, S., Analytic ideals and their applications . Annals of Pure and Applied Logic, vol. 99 (1999), pp. 5172.10.1016/S0168-0072(98)00051-7CrossRefGoogle Scholar
Steinhaus, H., Sur la convergence ordinaire et la convergence asymptotique . Colloquium Mathematicum, vol. 2 (1951), pp. 7374.Google Scholar
Szarek, S., A Banach space without a basis which has the bounded approximation property . Acta Mathematica, vol. 159 (1987), nos. 1–2, pp. 8198.CrossRefGoogle Scholar