On the sum of simultaneously proximinal sets

In this paper, we show that the sum of a compact convex subset and a simultaneously \(\tau\)-strongly proximinal convex subset (resp. simultaneously approximatively \(\tau\)-compact convex subset) of a Banach space X is simultaneously tau-strongly proximinal (resp. simultaneously approximatively \(\...

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Bibliographic Details
Published inarXiv.org
Main Authors Sun, Longfa, Sun, Yuqi, Zhang, Wen, Zheng, Zheming
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 27.02.2020
Subjects
Banach spaces
Subspaces
Topology
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Summary:In this paper, we show that the sum of a compact convex subset and a simultaneously \(\tau\)-strongly proximinal convex subset (resp. simultaneously approximatively \(\tau\)-compact convex subset) of a Banach space X is simultaneously tau-strongly proximinal (resp. simultaneously approximatively \(\tau\)-compact ), and the sum of weakly compact convex subset and a simultaneously approximatively weakly compact convex subset of X is still simultaneously approximatively weakly compact, where \(\tau\) is the norm or the weak topology. Moreover, some related results on the sum of simultaneously proximinal subspaces are presented.
ISSN:2331-8422