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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Nonexistence of continuous selections of the metric projection for a class of weak Chebyshev spaces


Author: Manfred Sommer
Journal: Trans. Amer. Math. Soc. 260 (1980), 403-409
MSC: Primary 41A65; Secondary 41A50, 41A52
DOI: https://doi.org/10.1090/S0002-9947-1980-0574787-3
MathSciNet review: 574787
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Abstract: The class $ \mathfrak{B}$ of all those n-dimensional weak Chebyshev subspaces of $ C\,[a,\,b]$ whose elements have no zero intervals is considered. It is shown that there does not exist any continuous selection of the metric projection for G if there is a nonzero g in G having at least $ n\, + \,1$ distinct zeros. Together with a recent result of Nürnberger-Sommer, the following characterization of continuous selections for $ \mathfrak{B}$ is valid: There exists a continuous selection of the metric projection for G in $ \mathfrak{B}$ if and only if each nonzero g in G has at most n distinct zeros.


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DOI: https://doi.org/10.1090/S0002-9947-1980-0574787-3
Keywords: Continuous selection, metric projection, weak Chebyshev spaces
Article copyright: © Copyright 1980 American Mathematical Society