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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Best approximation in $ L\sp 1(I,X)$


Authors: R. Khalil and F. Saidi
Journal: Proc. Amer. Math. Soc. 123 (1995), 183-190
MSC: Primary 41A65; Secondary 46E40
DOI: https://doi.org/10.1090/S0002-9939-1995-1223266-5
MathSciNet review: 1223266
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Abstract: Let X be a Banach space and G a closed subspace of X. The subspace G is called proximinal in X if for every $ x \in X$ there exists at least one $ y \in G$ such that $ \left\Vert {x - y} \right\Vert = d(x,G) = \inf \{ \left\Vert {x - z} \right\Vert:z \in G\} $.

It is an open problem whether $ {L^1}(I,G)$ is proximinal in $ {L^1}(I,X)$ if G is proximinal in X, where I is the unit interval with the Lebesgue measure.

In this paper, we prove the proximinality of $ {L^1}(I,G)$ in $ {L^1}(I,X)$ for a class of proximinal subspaces G in X.


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DOI: https://doi.org/10.1090/S0002-9939-1995-1223266-5
Article copyright: © Copyright 1995 American Mathematical Society