Weak convergence of bounded sequences

Rainwater, John

doi:10.1090/s0002-9939-1963-0155171-9

Cited by 32 publications

(30 citation statements)

References 1 publication

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“…also [15]). The case were B is the set of the extreme points of B X * is the well known Rainwater's theorem [21].…”

Section: A Proof Of James's Theorem In the Separable Casementioning

confidence: 99%

An extension of James's compactness theorem

Gasparis

2015

Journal of Functional Analysis

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Abstract. Let X and Y be Banach spaces and F ⊂ BY * . Endow Y with the topology τF of point-wise convergence on F . Assume that T : X * → Y is a bounded linear operator which is (w * , τF ) continuous. Assume further that every vector in the range of T attains its norm at some element of F (that is, for every x * ∈ X * there exists y * ∈ F such that T (x * ) = |y * (T x * )|). Then T is (w * , w) continuous. The proof relies on Rosenthal's ℓ1-theorem. As a corollary to the above result, one obtains an alternative proof of James's compactness theorem that a bounded subset K of a Banach space E is relatively weakly compact provided that each functional in E * attains its supremum on K.

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“…also [15]). The case were B is the set of the extreme points of B X * is the well known Rainwater's theorem [21].…”

Section: A Proof Of James's Theorem In the Separable Casementioning

confidence: 99%

An extension of James's compactness theorem

Gasparis

2015

Journal of Functional Analysis

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“…[25]) states that a bounded sequence in which is convergent to some ∈ under every functional from ex * is weakly convergent to . Later Simons (cf.…”

Section: A Sup-limsup-theoremmentioning

confidence: 99%

On Convergence with respect to an Ideal and a Family of Matrices

Hardtke

2014

International Journal of Analysis

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P. Das et al. recently introduced and studied the notions of strong -summability with respect to an Orlicz function andstatistical convergence, where is a nonnegative regular matrix and is an ideal on the set of natural numbers. In this paper, we will generalise these notions by replacing with a family of matrices and with a family of Orlicz functions or moduli and study the thus obtained convergence methods. We will also give an application in Banach space theory, presenting a generalisation of Simons' sup-limsup-theorem to the newly introduced convergence methods (for the case that the filter generated by the ideal has a countable base), continuing some of the author's previous work.

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“…Norming sets in Banach spaces have shown to be useful tools for studying several geometric and topological properties and have produced an important number of results in the literature ( [24,25,26]). Recall that a subset B of the dual E * of a Banach space E is said to be norming (or weak*-norming) if inf x∈S E sup x * ∈B | x, x * | > 0, where S E is the unit sphere of E.…”

Section: Introductionmentioning

confidence: 99%

Positively Norming Sets in Banach Function Spaces

Pérez

Tradacete

2013

The Quarterly Journal of Mathematics

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Abstract. The notion of positively norming set, a specific definition of norming type sets for Banach lattices, is analyzed. We show that the size of positively norming sets (in terms of compactness and order boundedness) is directly related to the existence of lattice copies of L 1 -spaces. As an application, we provide a version of Kadec-Pelczynski's dichotomy for order continuous Banach function spaces. A general description of positively norming sets using vector measure integration is also given.

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Weak convergence of bounded sequences

Cited by 32 publications

References 1 publication

An extension of James's compactness theorem

An extension of James's compactness theorem

On Convergence with respect to an Ideal and a Family of Matrices

Positively Norming Sets in Banach Function Spaces

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