Characterizations of Generalized Proximinal Subspaces in Real Banach Spaces

Let X be a real Banach space, C a closed bounded convex subset of X with the origin as an interior point, and p C the Minkowski functional generated by the set C . This paper is concerned with the problem of generalized best approximation with respect to p C . A property ( ε ∗ ) concerning a subspac...

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Bibliographic Details
Published in:Resultate der Mathematik 2019-09, Vol.74 (3), Article 88
Main Authors: Luo, Xian-Fa, Tao, Jicheng, Wei, Minxing
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:Let X be a real Banach space, C a closed bounded convex subset of X with the origin as an interior point, and p C the Minkowski functional generated by the set C . This paper is concerned with the problem of generalized best approximation with respect to p C . A property ( ε ∗ ) concerning a subspace of X ∗ is introduced to characterize generalized proximinal subspaces in X . A set C with feature as above in the space l 1 of absolutely summable sequences of real numbers and a continuous linear functional f on l 1 are constructed to show that each point in an open half space determined by the kernel of f admits a generalized best approximation from the kernel but each point in the other open half space does not.
ISSN:1422-6383
1420-9012
DOI:10.1007/s00025-019-1013-z