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Abstract  

A subspaceY of a Banach spaceX is called a Chebyshev one if for everyxX there exists a unique elementP Y(x) inY of best approximation. In this paper, necessary and sufficient conditions are obtained in order that certain classes of subspacesY of the Hardy spaceH 1=H 1 (|z|<1) be Chebyshev ones, and also the properties of the operatorP Y are studied. These results show that the theory of Chebyshev subspaces inH 1 differs sharply from the corresponding theory inL 1(C) of complex-valued functions defined and integrable on the unit circleC:|z|=1. For example, it is proved that inH 1 there exist sufficiently many Chebyshev subspaces of finite dimension or co-dimension (while inL 1(C) there are no Chebyshev subspaces of finite dimension or co-dimension). Besides, it turned out that the collection of the Chebyshev subspacesY with a linear operatorP Y inH 1 (in contrast toL 1(C)) is exhausted by that minimum which is necessary for any Banach space.

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geometrical conditions that implies the existence certain singular integral of Banach space-valued functions Proc. Conf. Harmonic Analysis Chicago 1981 Wads Worth Belmont 270 – 286 in honor of Antoni Zygmund

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Abstract  

The tensor integral of a vector valued function f: Ω → X with respect to a countably additive vector valued measure v: Σ → Y has been defined by Stefansson in [14] and he has investigated many of its properties. The integral is an element of the injective tensor product X

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Y. We study the Banach space L 1(v, X, Y) of all
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-integrable functions and discuss many properties of this space. We also study the space w-L 1(v, X, Y) of all weakly
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-integrable functions.

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Panyanak , B. , Nonexpansive set-valued mappings in metric and Banach spaces , Journal of Nonlinear and Convex Analysis , Vol. 8 , no. 1 ( 2007 ), pp. 35 – 45 . [10

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In this study, we define the spaces

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of double sequences whose Cesàro transforms are bounded, convergent in the Pringsheim’s sense, null in the Pringsheim’s sense, both convergent in the Pringsheim’s sense and bounded, regularly convergent and absolutely q-summable, respectively, and also examine some properties of those sequence spaces. Furthermore, we show that these sequence spaces are Banach spaces. We determine the alpha-dual of the space
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and the β(bp)-dual of the space
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, and β(ϑ)-dual of the space
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of double sequences, where ϑ, η ∈ {p, bp, r}. Finally, we characterize the classes (
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: C ϑ) and (μ:
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) for ϑ ∈ {p, bp, r} of four dimensional matrix transformations, where μ is any given space of double sequences.

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unconditional convergence of series in Banach spaces, Studia Math . 17 (1958), 151-164. MR 22 #5872 On bases and unconditional convergence of series in Banach spaces Studia Math

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Cauchy problem for differential-delay equations in a Banach space, Math. Nachr. 74 (1976), 173-190. MR 54 #10787 On the Cauchy problem for differential-delay equations in a Banach space Math

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.-Nagy, Gustafson) perturbation theorem for linear operators in Hilbert and Banach space, Acta Sci. Math. (Szeged) , 45(1–4 ) (1983), 201–211. Gustafson K. The RKNG (Rellich, Kato, Sz

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the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal. , 40 (1990), 11–19. Lakshmikantham V. Theorems about the existence

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Theory, Ankara University, 21 August–1 September, 2006. Kolk, E. , The statistical convergence in Banach spaces, Tartu Ül. Toimatised , 928 (1981), 41

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