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.
De Bernardi, C. A., Preti, A., Somaglia, J., A note on smooth rotund norms which are not midpoint locally uniformly rotund, <<JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS>>, 2025; 550 (2): N/A-N/A. [doi:10.1016/j.jmaa.2025.129544] [https://hdl.handle.net/10807/312159]
A note on smooth rotund norms which are not midpoint locally uniformly rotund
De Bernardi, Carlo Alberto
;
2025
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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