We prove that every separable infinite-dimensional Banach space admits a G & acirc;teaux 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 & eacute;chet 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.
A note on smooth rotund norms which are not midpoint locally uniformly rotund
De Bernardi, Carlo Alberto;Somaglia, Jacopo
2025-01-01
Abstract
We prove that every separable infinite-dimensional Banach space admits a G & acirc;teaux 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 & eacute;chet 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.File in questo prodotto:
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De Bernardi, Preti, Somaglia - A note on smooth rotund norms which are not midpoint locally uniformly rotund.pdf
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