We provide, in every infinite -dimensional separable Banach space, an average locally uniformly rotund (and hence rotund) Gateaux smooth renorming which is not locally uniformly rotund. This solves an open problem posed in a recent monograph by A.J. Guirao, V. Montesinos, and V. Zizler.
De Bernardi, C. A., Somaglia, J., ROTUND GÂTEAUX SMOOTH NORMS WHICH ARE NOT LOCALLY UNIFORMLY ROTUND, <<PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY>>, 2024; 152 (4): 1689-1701. [doi:10.1090/proc/16659] [https://hdl.handle.net/10807/270755]
ROTUND GÂTEAUX SMOOTH NORMS WHICH ARE NOT LOCALLY UNIFORMLY ROTUND
De Bernardi, Carlo Alberto
;
2024
Abstract
We provide, in every infinite -dimensional separable Banach space, an average locally uniformly rotund (and hence rotund) Gateaux smooth renorming which is not locally uniformly rotund. This solves an open problem posed in a recent monograph by A.J. Guirao, V. Montesinos, and V. Zizler.
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