We present a construction that enables one to find Banach spaces$X$whose sets$\operatorname{NA}(X)$of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently,$X$does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read [Banach spaces with no proximinal subspaces of codimension 2,Israel J. Math.(to appear)] and Rmoutil [Norm-attaining functionals need not contain 2-dimensional subspaces,J. Funct. Anal. 272(2017), 918–928]. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space$X$where the set$\operatorname{NA}(X)$for the original norm is not “too large”. The construction can be applied to every Banach space containing$c_{0}$and having a countable system of norming functionals, in particular, to separable Banach spaces containing$c_{0}$. We also provide some geometric properties of the norms we have constructed.

中文翻译：

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具有极端不可线集的范数获得函数的等价范数

我们提出了一种结构，使人们能够找到巴拿赫空间$X$谁的集合$\运营商名称{NA}(X)$范数获得泛函不包含二维子空间，因此，$X$不包含大于 1 的有限余维的近端子空间，扩展了 Read 最近提供的结果 [Banach spaces with no proximinal subspaces of codimension 2,以色列 J. 数学。（出现）] 和 Rmoutil [达到范数的泛函不需要包含二维子空间，J. 功能。肛门。 272(2017), 918–928]。粗略地说，我们为每个 Banach 空间构造了一个具有请求属性的等价重规范$X$集合在哪里$\运营商名称{NA}(X)$因为原来的规范不是“太大”。该构造可以应用于每个 Banach 空间，其中包含$c_{0}$并且具有规范泛函的可数系统，特别是可分离的 Banach 空间，其中包含$c_{0}$. 我们还提供了我们构建的规范的一些几何属性。