A remark on totally smooth renormings

Abstract

E. Oja, T. Viil, and D. Werner showed, in Totally smooth renormings, Archiv der Mathematik, 112, 3, (2019), 269-281, that a weakly compactly generated Banach space (X, • ) with the property that every linear functional on X has a unique Hahn-Banach extension to the bidual X * * (the so-called Phelps' property U in X * * , also known as the Hahn-Banach smoothness property) can be renormed to have the stronger property that for every subspace Y of X, every linear functional on Y has a unique Hahn-Banach extension to X * * (the so-called total smoothness property of the space). We mention here that this result holds in full generality -without any restriction on the space-and in a stronger form, thanks to a result of M. Raja, On dual locally uniformly rotund norms,