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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 50 Issue: 3

On the sum of simultaneously proximinal sets

Longfa Sun(North China Electric Power University)
Yuqi Sun(XIAMEN UNIVERSITY)
Wen Zhang(XIAMEN UNIVERSITY)0000-0001-9576-5644
Zheming Zheng(XIAMEN UNIVERSITY)
Pages
—
Publication Date
2021-06-07
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AI Summary & Key Takeaways

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⚡ TL;DR(Too Long; Didn't Read)

Moreover, some related results on the sum of simultaneously proximinal subspaces are presented.

📌 3 Key Takeaways

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Objective & Hypothesis: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.
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Key Findings & Data: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.
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Conclusion & Impact:Moreover, some related results on the sum of simultaneously proximinal subspaces are presented.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:simultaneous strong proximinality · simultaneous approximative compactness · compact sets

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.

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Cite This Article

Sun, L., Sun, Y., Zhang, W., Zheng, Z. (2021). On the sum of simultaneously proximinal sets. Hacettepe Journal of Mathematics and Statistics, Volume: 50 Issue: 3. https://doi.org/10.15672/hujms.696407

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