Some Generalizations of Locally Uniform Rotundity

Bandyopadhyay, P.; Huang, Da; Lin, Bor-Luh; Troyanski, Stanimir

doi:10.1006/jmaa.2000.7169

Cited by 22 publications

(16 citation statements)

References 14 publications

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“…From the results of [1], it follows that a Banach space X is τ -ALUR ⇔ every norm-attaining functional x * ∈ X * is a τ -strongly exposing functional. We refer to [1,2] for various characterizations and properties of such spaces.…”

Section: Introductionmentioning

confidence: 99%

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Proximinality in Banach spaces

Bandyopadhyay

Lin

et al. 2008

Journal of Mathematical Analysis and Applications

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In this paper, we study approximatively τ -compact and τ -strongly Chebyshev sets, where τ is the norm or the weak topology. We show that the metric projection onto τ -strongly Chebyshev sets are norm-τ continuous. We characterize approximatively τ -compact and τ -strongly Chebyshev hyperplanes and use them to characterize factor reflexive proximinal subspaces in τ -almost locally uniformly rotund spaces. We also prove some stability results on approximatively τ -compact and τ -strongly Chebyshev subspaces.

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Section: Introductionmentioning

confidence: 99%

“…In Section 3, we consider proximinality in τ -almost locally uniformly rotund (τ -ALUR) Banach spaces. These spaces were defined in [2] (see Definition 3.1). From the results of [1], it follows that a Banach space X is τ -ALUR ⇔ every norm-attaining functional x * ∈ X * is a τ -strongly exposing functional.…”

Section: Introductionmentioning

confidence: 99%

Proximinality in Banach spaces

Bandyopadhyay

Lin

et al. 2008

Journal of Mathematical Analysis and Applications

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“…They have been studied, e.g., in [1,7,16,18,19,23]. In particular, [7,19,20,22] contain applications to approximation theory.…”

Section: Definitions Notation and An Earlier Resultsmentioning

confidence: 99%

Geometric properties and continuity of the pre-duality mapping in Banach space

Zhang

Montesinos

Liu

et al. 2014

RACSAM

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Montesinos Santalucia, V.; Liu, CY.; Gong, WZ. (2015). Geometric properties and continuity of the pre-duality mapping in Banach space. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 109 (2) AbstractWe use the preduality mapping in proving characterizations of some geometric properties of Banach spaces. In particular, those include nearly strongly convexity, nearly uniform convexity -a property introduced by K. Goebel and T. Sekowski-, and nearly very convexity.

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“…Speaking in rotundity terms, to be a locally uniformly rotund point is the strongest thing that a vector of norm 1 can be. The paper [4] is an excellent reference for these kinds of properties. We prefer to omit the proof, which can be considered as an exercise.…”

Section: Geometry Of L 2 -Summand Vectorsmentioning

confidence: 99%

𝖫^{2}-summand vectors in Banach spaces

Aizpuru¹,

García-Pacheco

2006

Proc. Amer. Math. Soc.

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Abstract. The aim of this paper is to study the set L 2 X of all L 2 -summand vectors of a real Banach space X. We provide a characterization of L 2 -summand vectors in smooth real Banach spaces and a general decomposition theorem which shows that every real Banach space can be decomposed as an L 2 -sum of a Hilbert space and a Banach space without nontrivial L 2 -summand vectors. As a consequence, we generalize some results and we obtain intrinsic characterizations of real Hilbert spaces. BackgroundWe say that a Banach space X is smooth if every point x in X satisfies that there exists a unique f in X * such that f = x and f (x) = x 2 . If X is a smooth Banach space and x is in X, then we will denote this f by J X (x). The mapping J X : X −→ X * is usually called the duality mapping. Excellent books for consulting the duality mapping are [7] and [8].A closed subspace M of a real Banach space X is said to be an L 2 -summand subspace if there exists another closed subspace N of X verifying X = (M ⊕ N ) 2 ; in other words, m + n 2 = m 2 + n 2 for every m in M and every n in N . The linear projection π M of X onto M that fixes the elements of M and maps the elements of N to {0} is called the L 2 -summand projection of X onto M . Notice that N is uniquely determined, and hence so is π M . Good references for L 2 -summand subspaces are [2] and [3].A vector e of a real Banach space X is an L 2 -summand vector if Re is an L 2 -summand subspace. Furthermore, if e = 0, then there exists a functional e * in X * , which is called the L 2 -summand functional of e, such that e * = 1/ e , e * (e) = 1 and π Re (x) = e * (x) e for every x in X. We want to recall two relevant results about L 2 -summand vectors:(1

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Some Generalizations of Locally Uniform Rotundity

Abstract: Some new generalizations of locally uniform rotundity in Banach spaces are introduced and studied. ᮊ

Cited by 22 publications

References 14 publications

Proximinality in Banach spaces

Proximinality in Banach spaces

Geometric properties and continuity of the pre-duality mapping in Banach space

𝖫^{2}-summand vectors in Banach spaces

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