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.

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 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.

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

scholarly journals Sequences and bases in p-Banach spaces

1986 ◽  
Vol 34 (1) ◽  
pp. 87-92
Author(s):  
M. A. Ariño
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dimensional Subspace ◽  
Basic Sequence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Infinite Dimensional Subspace

Necessary and sufficient condition are given for an infinite dimensional subspace of a p-Banach space X with basis to contain a basic sequence which can be extended to a basis of X.

A Ramsey Theorem with an Application to Sequences in Banach Spaces

2012 ◽  
Vol 55 (2) ◽  
pp. 410-417
Author(s):  
Robert Service
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Ramsey Theory ◽  
Basic Sequence ◽  
Alternative Proof ◽  
Ramsey Theorem ◽  
Combinatorial Lemma

AbstractThe notion of a maximally conditional sequence is introduced for sequences in a Banach space. It is then proved using Ramsey theory that every basic sequence in a Banach space has a subsequence which is either an unconditional basic sequence or a maximally conditional sequence. An apparently novel, purely combinatorial lemma in the spirit of Galvin's theorem is used in the proof. An alternative proof of the dichotomy result for sequences in Banach spaces is also sketched, using the Galvin–Prikry theorem.

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.

scholarly journals Two remarks on the Franklin system

1986 ◽  
Vol 29 (3) ◽  
pp. 329-333 ◽  
Author(s):  
P. Wojtaszczyk
Keyword(s):  
Unconditional Basis ◽  
Unit Vector ◽  
Unit Vector Basis ◽  
Franklin System ◽  
Vector Basis

The aim of this note is to present two observations about the classical Franklin system. First we show that the Franklin system, when considered in the space generated by special atoms (as defined and studied by Soares de Souza in [11] and ]12]) is an unconditional basis equivalent to the unit vector basis in l1. In our second result we give conceptually simpler proofs and some extensions of the results of Bočkariov's [1] about the conjugate Franklin system.

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