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ON EQUIVALENCE RELATIONS GENERATED BY SCHAUDER BASES

Published online by Cambridge University Press:  09 January 2018

LONGYUN DING
LONGYUN DING*
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES AND LPMC NANKAI UNIVERSITY TIANJIN300071, P.R. CHINAE-mail: dinglongyun@gmail.com
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Abstract

In this article, a notion of Schauder equivalence relation ℝℕ/L is introduced, where L is a linear subspace of ℝℕ and the unit vectors form a Schauder basis of L. The main theorem is to show that the following conditions are equivalent:

(1) the unit vector basis is boundedly complete;

(2) L is a Fσ in ℝℕ;

(3) ℝℕ/L is Borel reducible to ℓ∞.

We show that any Schauder equivalence relation generalized by a basis of ℓ2 is Borel bireducible to ℝℕ/ℓ2 itself, but it is not true for bases of c0 or ℓ1. Furthermore, among all Schauder equivalence relations generated by sequences in c0, we find the minimum and the maximum elements with respect to Borel reducibility.

Keywords

Primary 03E1546B1546B45equivalence relationBorel reductionSchauder basis
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 4 , December 2017 , pp. 1459 - 1481
Copyright
Copyright © The Association for Symbolic Logic 2017 

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References

REFERENCES

Borodulin-Nadzieja, P., Farkas, B., and Plebanek, G., Representations of ideals in Polish groups in Banach spaces, this Journal, vol. 80 (2015), pp. 1268–1289.Google Scholar
Diestel, J., Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984.CrossRefGoogle Scholar
Ding, L., Borel reducibility and Hölder(α) embeddability between Banach spaces, this Journal, vol. 77 (2012), pp. 224–244.Google Scholar
Dougherty, R. and Hjorth, G., Reducibility and nonreducibility between ℓp equivalence relations. Transactions of the American Mathematical Society, vol. 351 (1999), pp. 1835–1844.CrossRefGoogle Scholar
Drewnowski, L. and Labuda, I., Ideals of subseries convergence and F-spaces. Archiv der Mathematik, vol. 108 (2017), pp. 55–64.CrossRefGoogle Scholar
Farah, I., Basis problem for turbulent actions I: Tsirelson submeasures. Annals of Pure and Applied Logic, vol. 108 (2001), pp. 189–203.CrossRefGoogle Scholar
Farah, I., Basis problem for turbulent actions II: c 0 -equalities. Proceedings of the London Mathematical Society, vol. 82 (2001), pp. 1–30.CrossRefGoogle Scholar
Gao, S., Invariant Descriptive Set Theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 293, CRC Press, Boca Raton, FL, 2008.Google Scholar
Gao, S. and Yin, Z., A note on equivalence relations l p (l q ). Mathematical Logic Quarterly, vol. 61 (2015), pp. 516–523.CrossRefGoogle Scholar
Gowers, W., An infinite Ramsey theorem and some Banach-space dichotomies. Annals of Mathematics, vol. 156 (2002), pp. 797–833.CrossRefGoogle Scholar
Hjorth, G., Actions by the classical Banach spaces, this Journal, vol. 65 (2000), pp. 392–420.Google Scholar
James, R. C., Bases and reflexivity of Banach spaces. Annals of Mathematics, vol. 52 (1950), pp. 518–527.CrossRefGoogle Scholar
Kadets, M. I. and Kadets, V. M., Series in Banach Spaces, Conditional and Unconditional Convergence, Birkhäuser, Basel, 1997.Google Scholar
Kanovei, V., Borel Equivalence Relations: Structure and Classification, University Lecture Series, vol. 44, American Mathematical Society, Providence, RI, 2008.Google Scholar
Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces I: Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92, Springer-Verlag, Berlin, 1977.CrossRefGoogle Scholar
Ma, X., On Schauder equivalence relations. Preprint, 2014, arXiv:1411.0327.Google Scholar
Mátrai, T., On ℓp-like equivalence relations. Real Analysis Exchange, vol. 34 (2008/09), pp. 377–412.CrossRefGoogle Scholar
Pelczynski, A. and Singer, I., On non-equivalent bases and conditional bases in Banach spaces. Studia Mathematica, vol. 25 (1964), pp. 5–25.CrossRefGoogle Scholar
Rosendal, C., Cofinal families of Borel equivalence relations and qusiorders, this Journal, vol. 70 (2005), pp. 1325–1340.Google Scholar
Yin, Z., Embeddings of p(ω)/fin into Borel equivalence relations between ℓp and ℓq , this Journal, vol. 80 (2015), pp. 917–939.Google Scholar
Zippin, M., A remark on bases and reflexivity in Banach spaces. Israel Journal of Mathematics, vol. 6 (1968), pp. 74–79.CrossRefGoogle Scholar