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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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