A note on smooth rotund norms which are not midpoint locally uniformly rotund

2024/02/21 by De Bernardi, Carlo Alberto, Preti, Alessandro, Somaglia, Jacopo
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2402.13869

Abstract

We prove that every separable infinite-dimensional Banach space admits a Gâteaux 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 Fréchet 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.

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