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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 TIANJIN 300071, P.R. CHINA E-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 F σ in ℝℕ;

(3) ℝℕ/L is Borel reducible to ℓ ∞.

We show that any Schauder equivalence relation generalized by a basis of ℓ 2 is Borel bireducible to ℝℕ/ℓ 2 itself, but it is not true for bases of c 0 or ℓ 1. Furthermore, among all Schauder equivalence relations generated by sequences in c 0, we find the minimum and the maximum elements with respect to Borel reducibility.

Keywords

Primary 03E1546B1546B45equivalence relationBorel reductionSchauder basis

Information

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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