- Hacettepe Journal of Mathematics and Statistics
- Vol: 50 Issue: 3
- On the sum of simultaneously proximinal sets
On the sum of simultaneously proximinal sets
Authors : Longfa Sun, Yuqi Sun, Wen Zhang, Zheming Zheng
Pages : 668-677
Doi:10.15672/hujms.696407
View : 10 | Download : 5
Publication Date : 2021-06-07
Article Type : Research
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