scholarly journals ON EQUIVALENCE RELATIONS GENERATED BY SCHAUDER BASES

2017 ◽  
Vol 82 (4) ◽  
pp. 1459-1481
Author(s):  
LONGYUN DING
Keyword(s):  
Equivalence Relation ◽  
Linear Subspace ◽  
Unit Vector ◽  
Unit Vector Basis ◽  
Schauder Basis ◽  
Equivalence Relations ◽  
Vector Basis ◽  
Schauder Bases ◽  
Borel Reducibility ◽  
Unit Vectors

AbstractIn 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.

Banded perturbations of the unit vector basis in some sequence spaces

1978 ◽  
Vol 29 (4) ◽  
pp. 389-392
Author(s):  
Alfred D. Andrew ◽  
Stephen Demko
Keyword(s):  
Unit Vector ◽  
Sequence Spaces ◽  
Unit Vector Basis ◽  
Vector Basis

A stability property of the unit vector basis ofl p

1975 ◽  
Vol 20 (3-4) ◽  
pp. 216-227 ◽  
Author(s):  
A. Szankowski ◽  
M. Zippin
Keyword(s):  
Stability Property ◽  
Unit Vector ◽  
Unit Vector Basis ◽  
Vector Basis

Bi-Borel reducibility of essentially countable Borel equivalence relations

2005 ◽  
Vol 70 (3) ◽  
pp. 979-992 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Borel Space ◽  
Borel Equivalence Relations ◽  
Borel Equivalence Relation ◽  
Standard Borel Space ◽  
Borel Reducibility ◽  
New Feature ◽  
Borel Probability ◽  
Image Position

This note answers a questions from [2] by showing that considered up to Borel reducibility, there are more essentially countable Borel equivalence relations than countable Borel equivalence relations. Namely:Theorem 0.1. There is an essentially countable Borel equivalence relation E such that for no countable Borel equivalence relation F (on a standard Borel space) do we haveThe proof of the result is short. It does however require an extensive rear guard campaign to extract from the techniques of [1] the followingMessy Fact 0.2. There are countable Borel equivalence relationssuch that:(i) eachExis defined on a standard Borel probability space (Xx, μx); each Ex is μx-invariant and μx-ergodic;(ii) forx1 ≠ x2 and A μxι -conull, we haveExι/Anot Borel reducible toEx2;(iii) if f: Xx → Xxis a measurable reduction ofExto itself then(iv)is a standard Borel space on which the projection functionis Borel and the equivalence relation Ê given byif and only ifx = x′ andzExz′ is Borel;(V)is Borel.We first prove the theorem granted this messy fact. We then prove the fact.(iv) and (v) are messy and unpleasant to state precisely, but are intended to express the idea that we have an effective parameterization of countable Borel equivalence relations by points in a standard Borel space. Examples along these lines appear already in the Adams-Kechris constructions; the new feature is (iii).Simon Thomas has pointed out to me that in light of theorem 4.4 [5] the Gefter-Golodets examples of section 5 [5] also satisfy the conclusion of 0.2.

Complemented Copies of ℓ1 and Pełczyński's Property (V*) in Bochner Function Spaces

1996 ◽  
Vol 48 (3) ◽  
pp. 625-640 ◽  
Author(s):  
Narcisse Randrianantoanina
Keyword(s):  
Banach Space ◽  
Unit Vector ◽  
Function Spaces ◽  
Bounded Sequence ◽  
Unit Vector Basis ◽  
Vector Basis

AbstractLet X be a Banach space and (fn)n be a bounded sequence in L1(X). We prove a complemented version of the celebrated Talagrand's dichotomy, i.e., we show that if (en)n denotes the unit vector basis of c0, there exists a sequence gn ∈ conv(fn, fn+1,...) such that for almost every ω, either the sequence (gn(ω) ⊗ en) is weakly Cauchy in or it is equivalent to the unit vector basis of ℓ1. We then get a criterion for a bounded sequence to contain a subsequence equivalent to a complemented copy of ℓ1 in L1(X). As an application, we show that for a Banach space X, the space L1(X) has Pełczyńiski's property (V*) if and only if X does.

scholarly journals Cofinal families of Borel equivalence relations and quasiorders

2005 ◽  
Vol 70 (4) ◽  
pp. 1325-1340 ◽  
Author(s):  
Christian Rosendal
Keyword(s):  
Equivalence Relation ◽  
Metric Spaces ◽  
Equivalence Relations ◽  
Borel Equivalence Relations ◽  
Compact Metric Spaces ◽  
Borel Equivalence Relation ◽  
Borel Ideal ◽  
Borel Reducibility

AbstractFamilies of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering. ≤B, are constructed. There is an analytic ideal on ω generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spaces, are shown to be Kσ complete.

scholarly journals On Perfectly Homogeneous Bases in Quasi-Banach Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-7
Author(s):  
F. Albiac ◽  
C. Leránoz
Keyword(s):  
Banach Spaces ◽  
Classical Result ◽  
Unit Vector ◽  
Unit Vector Basis ◽  
Vector Basis ◽  
Symmetric Bases ◽  
Special Case

For0<p<∞the unit vector basis ofℓphas the property of perfect homogeneity: it is equivalent to all its normalized block basic sequences, that is, perfectly homogeneous bases are a special case of symmetric bases. For Banach spaces, a classical result of Zippin (1966) proved that perfectly homogeneous bases are equivalent to either the canonicalc0-basis or the canonicalℓp-basis for some1≤p<∞. In this note, we show that (a relaxed form of) perfect homogeneity characterizes the unit vector bases ofℓpfor0<p<1as well amongst bases in nonlocally convex quasi-Banach spaces.

Analytic equivalence relations and bi-embeddability

2011 ◽  
Vol 76 (1) ◽  
pp. 243-266 ◽  
Author(s):  
Sy-David Friedman ◽  
Luca Motto Ros
Keyword(s):  
Banach Spaces ◽  
Equivalence Relation ◽  
Equivalence Relations ◽  
Topological Spaces ◽  
Polish Spaces ◽  
Borel Reducibility ◽  
Natural Classes ◽  
Strong Contrast

AbstractLouveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of ) is far from complete (see [5, 2]).In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and bi-embeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of , but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids.

Perfectly Homogeneous Bases in Banach Spaces

1975 ◽  
Vol 18 (1) ◽  
pp. 137-140 ◽  
Author(s):  
P. G. Casazza ◽  
Bor-Luh Lin
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Unit Vector ◽  
Basic Sequence ◽  
Unit Vector Basis ◽  
Vector Basis ◽  
Homogeneous Basis

A bounded basis {xn} of a Banach space X is called perfectly homogeneous if every bounded block basic sequence {yn} of {xn} is equivalent to {xn}. By a result of M. Zippin [4], a basis in a Banach space is perfectly homogeneous if and only if it is equivalent to the unit vector basis of c0 or lp, 1 ≤ p < + ∞. A basis {xn} of a Banach space X is called symmetric, if every permutation {xσ(n)} of {xn} is a basis of X, equivalent to the basis {xn}. It is clear that every perfectly homogeneous basis is symmetric.

Sign in / Sign up

Export Citation Format

Share Document