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 \((X,\Vert \cdot \Vert )\) 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, Israel Journal of Mathematics 129 (2002), 77–91.
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Acknowledgements
Supported by AEI/FEDER (project MTM2017-83262-C2-2-P of Ministerio de Economía y Competitividad), by Fundación Séneca, Región de Murcia (Grant 19368/PI/14), and Universitat Politècnica de València (A. J. Guirao). Supported by AEI/FEDER (project MTM2017-83262-C2-1-P of Ministerio de Economía y Competitividad) and Universitat Politècnica de València (V. Montesinos). We thank the referees for their work, that neatly improved the original version of this note to its final form.
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Cobollo, C., Guirao, A.J. & Montesinos, V. A remark on totally smooth renormings. RACSAM 114, 103 (2020). https://doi.org/10.1007/s13398-020-00831-5
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DOI: https://doi.org/10.1007/s13398-020-00831-5