标题： 博赫纳可积函数空间中最佳逼近性质的各种概念 显示英文标题

标题： Various notions of best approximation property in spaces of Bochner integrable functions

摘要： 我们得出，对于可分的逼近子空间$Y$ of$X$，$Y$是强逼近的（强球逼近的）当且仅当对于$1\leq p< \infty$，$L_p(I,Y)$在$L_p(I,X)$中是强逼近的（强球逼近的）。 $p=\infty$的情况由$Y$在$X$中的更强假设（一致可逼近性）得出。 观察到对于 $Y$ 中的可分近似子空间 $X$，当且仅当 $L_p(I,Y)$ 在 $L_p(I,X)$ 中是球近似时， $Y$ 在 $X$ 中是球近似，其中 $1\leq p\leq\infty$。 我们的观察还包括这样一个事实，对于任何（强）可逼近子空间$Y$的$X$，如果$Y$的每一个可分子空间在$X$中是球（强）可逼近的，那么$L_p(I,Y)$在$L_p(I,X)$中对于$1\leq p<\infty$是球（强）可逼近的。 我们引入了巴拿赫空间中闭凸集的均匀逼近性的概念，该概念在\cite{LZ}中被错误定义。给出了具有此性质的几个例子，即：巴拿赫空间中的任何$U$子空间，具有$3.2.I.P$的空间的闭单位球$B_X$，具有$3.2.I.P.$的空间的任何 M-理想的空间的闭单位球都是均匀逼近的。给出了具有此性质的新一类例子。 显示英文摘要

摘要： We derive that for a separable proximinal subspace $Y$ of $X$, $Y$ is strongly proximinal (strongly ball proximinal) if and only if for $1\leq p< \infty$, $L_p(I,Y)$ is strongly proximinal (strongly ball proximinal) in $L_p(I,X)$. Case for $p=\infty$ follows from stronger assumption on $Y$ in $X$ (uniform proximinality). It is observed that for a separable proximinal subspace $Y$ in $X$, $Y$ is ball proximinal in $X$ if and only if $L_p(I,Y)$ is ball proximinal in $L_p(I,X)$ for $1\leq p\leq\infty$. Our observations also include the fact that for any (strongly) proximinal subspace $Y$ of $X$, if every separable subspace of $Y$ is ball (strongly) proximinal in $X$ then $L_p(I,Y)$ is ball (strongly) proximinal in $L_p(I,X)$ for $1\leq p<\infty$. We introduce the notion of uniform proximinality of a closed convex set in a Banach space, which is wrongly defined in \cite{LZ}. Several examples are given having this property, viz. any $U$-subspace of a Banach space, closed unit ball $B_X$ of a space with $3.2.I.P$, closed unit ball of any M-ideal of a space with $3.2.I.P.$ are uniformly proximinal. A new class of examples are given having this property.