A remark on totally smooth renormings

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 $(X,\mathrm{\parallel }\cdot \mathrm{\parallel })$ with the property that every linear functional on $X$ has a unique Hahn–Banach extension to the bidual $X^{\ast \ast }$ (the so-called Phelps’ property U in $X^{\ast \ast }$, 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 $X^{\ast \ast }$ (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, Israel Journal of Mathematics 129 (2002), 77–91].