Library of Congress Cataloging-in-Publieation Data Funetional analysis andinfinite-dimensional geometry / Mariän Fabian . . . [et al.). p. cm . -(CMS books in mathematies ; 8) Ineludes bibliographical references andindex. I. Funetional analysis. 2. Banaeh spaces. I. Fabian, Mariän J. 1I. Series. QA320 .F793 2001 5I5'.7--de2 I 00-053773

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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,