A Class of Hereditarily l p Banach Spaces Without Schur Property

2017 ◽  
Vol 42 (1) ◽  
pp. 1-4
Author(s):  
Alireza Ahmadi
Keyword(s):  
Banach Spaces ◽  
Schur Property

scholarly journals Strongly exposed points and a characterization of l1 (Γ) by the Schur property

1987 ◽  
Vol 30 (3) ◽  
pp. 397-400 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Spaces ◽  
Banach Lattice ◽  
Positive Cone ◽  
Schur Property ◽  
Positive Elements ◽  
Strongly Exposed Points ◽  
Ordered Banach Spaces ◽  
Convex Subsets ◽  
Quasi Interior

In this paper we study the existence of strongly exposed points in unbounded closed and convex subsets of the positive cone of ordered Banach spaces and we prove the following characterization for the space l1(Γ): A Banach lattice X is order-isomorphic to l1(Γ) iff X has the Schur property and X* has quasi-interior positive elements.

Lifting results for sequences in Banach spaces

1989 ◽  
Vol 105 (1) ◽  
pp. 117-121 ◽  
Author(s):  
M. Gonzalez ◽  
V. M. Onieva
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Cauchy Sequence ◽  
Convergent Sequence ◽  
Convergent Subsequence ◽  
Bounded Sequence ◽  
Convergence Properties ◽  
Schur Property ◽  
Finite Dimensional

Several important classes of Banach spaces are characterized by means of convergence properties of sequences. For example, if X is a Banach space, then X belongs to the class Nl1 of spaces without copies of l1, the class R of reflexive spaces or the class F of finite-dimensional spaces if and only if each bounded sequence has respectively a weakly Cauchy (w-Cauchy), weakly convergent (w-convergent) or convergent subsequence. Similarly X is in the class WSC of weakly sequentially complete spaces, or the class SCH of spaces with the Schur property if and only if each w-Cauchy sequence is w-convergent, or convergent, respectively; note that X ∈ SCH if and only if each w-convergent sequence of X is convergent (see [12], p. 47).

On the $p$ -Schur property of Banach spaces

2018 ◽  
Vol 9 (1) ◽  
pp. 123-136 ◽  
Author(s):  
Mohammad B. Dehghani ◽  
S. Mohammad Moshtaghioun
Keyword(s):  
Banach Spaces ◽  
Schur Property

A ξ-weak Grothendieck compactness principle

2021 ◽  
pp. 1-16
Author(s):  
KEVIN BEANLAND ◽  
RYAN M. CAUSEY
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Banach Spaces ◽  
Compact Set ◽  
Closed Convex Hull ◽  
Compact Sets ◽  
Weakly Compact ◽  
Schur Property ◽  
Compactness Principle ◽  
Weakly Compact Sets

Abstract For 0 ≤ ξ ≤ ω1, we define the notion of ξ-weakly precompact and ξ-weakly compact sets in Banach spaces and prove that a set is ξ-weakly precompact if and only if its weak closure is ξ-weakly compact. We prove a quantified version of Grothendieck’s compactness principle and the characterisation of Schur spaces obtained in [7] and [9]. For 0 ≤ ξ ≤ ω1, we prove that a Banach space X has the ξ-Schur property if and only if every ξ-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence. The ξ = 0 and ξ= ω1 cases of this theorem are the theorems of Grothendieck and [7], [9], respectively.

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