Some subspaces simultaneously proximinal

Tijani Pakhrou

doi:10.1108/03684921211213142

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Kybernetes

ISSN: 0368-492X

Online from: 1972

Subject Area: Electrical & Electronic Engineering

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Some subspaces simultaneously proximinal

Document Information:
Title:	Some subspaces simultaneously proximinal
Author(s):	Tijani Pakhrou, (Department of Mathematical Analysis, Faculty of Sciences, University of Alicante, Alicante, Spain)
Citation:	Tijani Pakhrou, (2012) "Some subspaces simultaneously proximinal", Kybernetes, Vol. 41 Iss: 1/2, pp.108 - 115
Keywords:	Approximations and expansions, Best approximation, Chebyshev systems, Geometry and structure of normed linear spaces, Mathematics, Simultaneous approximation
Article type:	Research paper
DOI:	10.1108/03684921211213142 (Permanent URL)
Publisher:	Emerald Group Publishing Limited
Acknowledgements:	Supported in part by MTM2008-02652 Grant MTM of the Spanish MICINN.
Abstract:	Purpose – In this paper the aim is to present some subspace simultaneously proximinal in the Banach space L1(µ,?X) of X-valued Bochner µ-integrable functions. Design/methodology/approach – By lower semicontinuity and compactness the existence of best simultaneous approximation is obtained. Findings – If Y is a reflexive subspace of a Banach space X, then L1(µ,?Y) is simultaneously proximinal in L1(µ,?X). Furthermore, if X is reflexive and µ0 is the restriction of µ to a sub-s-algebra, then L1(µ0,?X) is simultaneously proximinal in L1(µ,?X). Practical implications – Given a finite number of points in the Banach space X, is about finding a point in the subspace Y?X that comes close to all this points. Originality/value – By the property of reflexivity two types subspaces simultaneously proximinal in L1(µ,?X) are obtained.