On the sum of simultaneously proximinal sets

Year 2021, Volume: 50 Issue: 3, 668 - 677, 07.06.2021

Longfa Sun Yuqi Sun Wen Zhang Zheming Zheng

https://doi.org/10.15672/hujms.696407

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 a 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.

Keywords

simultaneous strong proximinality, simultaneous approximative compactness, compact sets, Banach space

Supporting Institution

Fundamental Research Funds for the Central Universities, National Natural Science Foundation of China.

Project Number

2019MS121, 11731010.

Thanks

The authors were supported by the Fundamental Research Funds for the Central Universities 2019MS121 and the National Natural Science Foundation of China (no. 11731010 and 12071388).

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