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arXiv · 2609.07935

Weakly LUR norms and the Schur property

Szymon Draga·Tomasz Kania
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Abstract

A separable real or complex Banach space fails the Schur property if and only if it admits an equivalent G\^ateaux 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 G\^ateaux smooth whenever the prescribed norm is G\^ateaux smooth; Fr\'echet 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.

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Szymon Draga, Tomasz Kania. 2026-09-07. Weakly LUR norms and the Schur property. https://arxiv.org/abs/2609.07935

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