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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Best approximation in $L^ 1(I,X)$
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by R. Khalil and F. Saidi PDF
Proc. Amer. Math. Soc. 123 (1995), 183-190 Request permission

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 $\left \| {x - y} \right \| = d(x,G) = \inf \{ \left \| {x - z} \right \|:z \in G\}$. 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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • 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