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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2021 / Cilt: 50 - Sayı: 3

On the sum of simultaneously proximinal sets

Longfa SUN
Yuqi SUN
Wen ZHANG
Zheming ZHENG
Sayfa
668–677
Yayın Tarihi
2022-10-18
DOI
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Moreover, some related results on the sum of simultaneously proximinal subspaces are presented.

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Amaç & Hipotez: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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Temel Bulgular & Veri: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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Sonuç & Katkı:Moreover, some related results on the sum of simultaneously proximinal subspaces are presented.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler: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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Bu Makaleye Atıf Yap

SUN, L., SUN, Y., ZHANG, W., ZHENG, Z. (2022). On the sum of simultaneously proximinal sets. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 3, 668–677. https://ajindex.com/articles/529531

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