Stability of ball proximinality

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• 作者：Pei-Kee Lin^a ; ^{pklin@memphis.edu" class="auth_mail} ; Wen Zhang^b ; ^{wenzhang@xmu.edu.cn" class="auth_mail} ; Bentuo Zheng^a ; ^{bzheng@memphis.edu" class="auth_mail}
• 关键词：46B20 ; 41A50
• 刊名：Journal of Approximation Theory
• 出版年：July 2014
• 年：2014
• 卷：183
• 期：Complete
• 页码：72-81
• 全文大小：195 K

文摘

In this paper, we show that if E$E$ is an order continuous Köthe function space and Y$Y$ is a separable subspace of X$X$, then E(Y)$E\left(Y\right)$ is ball proximinal in E(X)$E\left(X\right)$ if and only if Y$Y$ is ball proximinal in X$X$. As a consequence, E(Y)$E\left(Y\right)$ is proximinal in 1a200" title="Click to view the MathML source">E(X)$E\left(X\right)$ if and only if Y$Y$ is proximinal in X$X$. This solves an open problem of Bandyopadhyay, Lin and Rao. It is also shown that if E$E$ is a Banach lattice with a 1$1$-unconditional basis and for each n$n$, Y_n$Y_n$ is a subspace of X_n$X_n$, then (⊕Y_n)_E${(\oplus Y_n)}_E$ is ball proximinal in (⊕X_n)_E${(\oplus X_n)}_E$ if and only if each Y_n$Y_n$ is ball proximinal in X_n$X_n$.