Abstract
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 \((\varepsilon _*)\) 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.
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Acknowledgements
The authors would like to thank the referees for their careful reading and valuable suggestions. In particular, for the proofs of the implications (ii)\(\Rightarrow \)(iii) and (iii)\(\Rightarrow \)(iv) in Theorem 1, the authors adopted the methods of one of them, which simplified the corresponding presentation. The research of the first two authors is supported in part by the Natural Sciences Foundation of Zhejiang Province (Grant No. LY16A010009).
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Luo, XF., Tao, J. & Wei, M. Characterizations of Generalized Proximinal Subspaces in Real Banach Spaces. Results Math 74, 88 (2019). https://doi.org/10.1007/s00025-019-1013-z
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DOI: https://doi.org/10.1007/s00025-019-1013-z