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arXiv · 2002.11961

On the sum of simultaneously proximinal sets

Longfa Sun·Yuqi Sun·Wen Zhang·Zheming Zheng
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Abstract

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.

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Longfa Sun, Yuqi Sun, Wen Zhang, Zheming Zheng. 2020-02-27. On the sum of simultaneously proximinal sets. https://arxiv.org/abs/2002.11961

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