发表机构
Czech Academy of Sciences; Jagiellonian University（捷克科学院; 雅盖隆大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明可分Banach空间不满足Schur性质当且仅当存在等价Gâteaux光滑弱局部一致凸非中点局部一致凸范数，并给出任意空间的充要条件及直接构造。
AI 中文摘要
一个可分实或复 Banach 空间不满足 Schur 性质，当且仅当它存在一个等价的 Gâteaux 光滑、弱局部一致凸但不中点局部一致凸的范数。该构造通过一个弱紧集扩大 LUR 单位球，并添加一个带权重的 Hilbert 空间项。所得范数可以任意接近任何给定的等价 LUR 范数，并且当给定范数是 Gâteaux 光滑时，所得范数也是 Gâteaux 光滑的；Fréchet 光滑性同样得以保持。在每个可分非 Schur 空间上，弱局部一致凸但不中点局部一致凸的范数在所有等价范数中是稠密的。对于任意实或复 Banach 空间，这样的重赋范存在当且仅当该空间是 LUR 可重赋范的且不满足 Schur 性质。我们对这最后一项断言给出了直接构造。
英文摘要
A separable real or complex Banach space fails the Schur property if and only if it admits an equivalent Gâteaux smooth, weakly locally uniformly rotund norm which is not midpoint locally uniformly rotund. The construction enlarges an LUR unit ball by a weakly compact set and adds a weighted Hilbert-space term. The resulting norms can be chosen arbitrarily close to any prescribed equivalent LUR norm, and are Gâteaux smooth whenever the prescribed norm is Gâteaux smooth; Fréchet smoothness is preserved as well. Weakly locally uniformly rotund norms which are not midpoint locally uniformly rotund are dense among all equivalent norms on each separable non-Schur space. For arbitrary real or complex Banach spaces, such a renorming exists exactly when the space is LUR-renormable and fails the Schur property. We give a direct construction for this last assertion.
