Weakly LUR norms and the Schur property
Abstract
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.
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BibTeX
@misc{draga2026weakly,
title = {{Weakly LUR norms and the Schur property}},
author = {Szymon Draga and Tomasz Kania},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07935},
eprint = {2609.07935},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07935},
}
RIS
TY - UNPB
TI - Weakly LUR norms and the Schur property
AU - Szymon Draga
AU - Tomasz Kania
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07935
UR - https://arxiv.org/abs/2609.07935
AB - 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.
ER -