A note on smooth rotund norms which are not midpoint locally uniformly rotund
Functional Analysis
2025-04-08 v2
Abstract
We prove that every separable infinite-dimensional Banach space admits a G\^ateaux smooth and rotund norm which is not midpoint locally uniformly rotund. Moreover, by using a similar technique, we provide in every infinite-dimensional Banach space with separable dual a Fr\'echet smooth and weakly uniformly rotund norm which is not midpoint locally uniformly rotund. These two results provide a positive answer to some open problems by A. J. Guirao, V. Montesinos, and V. Zizler.
Keywords
Cite
@article{arxiv.2402.13869,
title = {A note on smooth rotund norms which are not midpoint locally uniformly rotund},
author = {Carlo Alberto De Bernardi and Alessandro Preti and Jacopo Somaglia},
journal= {arXiv preprint arXiv:2402.13869},
year = {2025}
}