Banach Spaces of Vector-Valued Functions

Cembranos, Pilar; Mendoza, José

doi:10.1007/bfb0096765

Cited by 107 publications

(87 citation statements)

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“…Then according to the Radon-Nikodym type theorem (see [49,Theorem 1.5.3]) there exists a weakmeasurable function g y 0 W X ! E 0 such that kg y 0 .…”

Section: Radon Extension Of Operators Onmentioning

confidence: 99%

A Riesz representation theory for completely regular Hausdorff spaces and its applications

Nowak

2016

Open Mathematics

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Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let C b .X; E/ be the space of all E-valued bounded, continuous functions on X , equipped with the strict topologyˇ. We develop the Riemman-Stieltjes-type integral representation theory of .ˇ; k k F /-continuous operators T W C b .X; E/ ! F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded .ˇ; k k F /-continuous operators T W C b .X; E/ ! F . As an application, we study .ˇ; k k F /-continuous weakly compact and unconditionally converging operators T W C b .X; E/ ! F . In particular, we establish the relationship between these operators and the corresponding Borel operator measures given by the Riesz representation theorem. We obtain that if X is a k-space and E is reflexive, then .C b .X; E/;ˇ/ has the V property of Pełczynski.

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“…Then according to the Radon-Nikodym type theorem (see [49,Theorem 1.5.3]) there exists a weakmeasurable function g y 0 W X ! E 0 such that kg y 0 .…”

Section: Radon Extension Of Operators Onmentioning

confidence: 99%

A Riesz representation theory for completely regular Hausdorff spaces and its applications

Nowak

2016

Open Mathematics

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“…We shall say that two functions f, g ∈ H 1 (μ, X) are μ-equivalent if there is a μ-zero set N such that f (ω) = g(ω) for all ω ∈ \ N and we shall denote by H 1 (μ, X) the set of all classes of μ-equivalent functions. By L ∞ w * (μ, X * ) we shall design the linear space over ‫ދ‬ of all μ-essentially bounded functions ϕ : → X * which are weak* measurable whereas bvca μ ( , X) will represent the linear subspace of bvca( , X) of all those measures F for which there is some a > 0 (which depends on F) such that F(E) ≤ a μ(E) for each E ∈ , [2]. DEFINITION 2.1.…”

Section: Banach Spaces With Property (M) If ( μ)mentioning

confidence: 99%

bvca(Σ, X) REVISITED

Ferrando

2010

Glasgow Math. J.

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Abstract. Assuming that ( , ) is a measurable space and X is a Banach space we provide a quite general sufficient condition on X for bvca( , X) (the Banach space of all X-valued countably additive measures of bounded variation equipped with the variation norm) to contain a copy of c 0 if and only if X does. Some well-known results on this topic are straightforward consequences of our main theorem.2010 Mathematics Subject Classification. 28B05, 46B03.1. Preliminaries. Throughout this paper X will be a Banach space over the field ‫ދ‬ of real or complex numbers. Our notation is standard [3,4]. If ( , ) is a measurable space, ca( , X) denotes the Banach space over ‫ދ‬ of all X-valued countably additive measures F on provided with the semivariation norm F and bvca( , X) stand for the Banach space of all X-valued countably additive measures F of bounded variation on equipped with the variation norm |F|. We represent by ca + ( ) the set of all positive and finite measures defined on . If ( , , μ) is a finite measure space, recall that a weakly μ-measurable function f : → X is said to be Dunford integrable if x * f ∈ L 1 (μ) for every x * ∈ X * . If f is Dunford integrable and E ∈ the map x * → E x * f dμ, denoted by (D) E f dμ, is a continuous linear form on X * . If (D) E f dμ ∈ X for each E ∈ then f is called Pettis integrable and one writes (P) E f dμ instead of (D) E f dμ. A strongly μ-measurable function f : → X is said to be Bochner integrable if f (ω) dμ(ω) < ∞. As usual we denote by L 1 (μ, X) the Banach space of all (equivalence classes of) μ-Bochner integrable functions equipped with the normRecall that a series ∞ n=1 x n in X is said to be weakly unconditionally Cauchy (wuC) if+ ( ) is purely atomic, then ca( , X) contains a copy of c 0 or ∞ if and only if X contains, respectively, a copy of c 0 or ∞ [5]. Assuming that X has the Radon-Nikodym property with respect to each μ ∈ ca + ( ), then bvca( , X) contains a copy of c 0 or ∞ if and only if X does [7]. As a consequence, if each μ ∈ ca + ( ) is purely atomic then bvca( , X) contains a copy of c 0 or ∞ if and only if X contains, respectively, a copy of c 0 or ∞ . If there exists a nonzero atomless measure μ ∈ ca + ( ), the latter statement is no longer true [11]. However, if the range space of the measures is a dual Banach space X * , then bvca( , X * ) contains a copy of c 0 or ∞ if and only if X * does [10].

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“…The reader may peruse [5,9], where it is proved that if both and E have the Dunford-Pettis property (resp. the hereditary Dunford-Pettis property) then so does ðX Þ or [13], devoted to the stability of the properties of containing l p or c 0 .…”

Section: Elementary Stability Propertiesmentioning

confidence: 99%