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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach space theory: the basis of linear and nonlinear analysis. Springer, New York (2011)
Fabian, M., Montesinos, V., Zizler, V.: Smoothness in Banach spaces. Selected problems. Rev. R. Acad. Cien. Ser. A. Mat. RACSAM. 100(1–2), 101–125 (2006)
Ferrari, S., Orihuela, J., Raja, M.: Generalized metric properties of spheres and renorming of Banach spaces. Rev. R. Acad. Cienc. Exactas Fis. Natl. Ser. A Math. RACSAM. 113, 2655–2663 (2019)
Foguel, S.R.: On a theorem by A. E. Taylor. Proc. Amer. Math. Soc. 9, 325 (1958)
Godefroy, G.: Points de Namioka, espaces normants, applications à la théorie isométrique de la dualité. Israel J. Math. 38, 209–220 (1981)
Guirao, A.J., Montesinos, V., Zizler, V.: Open Problems in the geometry and analysis of Banach spaces. Springer International Pub, Switzerland (2016)
Harmand, P., Werner, D., Werner, W.: M-ideals in Banach spaces and Banach algebras. Lecture notes in math, vol. 1547. Springer, Berlin (1993)
Haydon, R.: Locally uniformly rotund norms in Banach spaces and their duals. J. Funct. Anal. 254, 2023–2039 (2008)
Oja, E., Viil, T., Werner, D.: Totally smooth renormings. Archiv. der. Mathematik. 112(3), 269–281 (2019)
Phelps, R.R.: Uniqueness of Hahn–Banach extensions and unique best approximation. Trans. Amer. Math. Soc. 95, 238–255 (1960)
Raja, M.: On dual locally uniformly rotund norms. Israel J. Math. 129, 77–91 (2002)
Smith, R.J., Troyanski, S.L.: Renormings of \(C(K)\) spaces. Rev. R. Acad. Cienc. Exactas Fís. Natl. Ser. A Math. RACSAM 104(2), 375–412 (2010)
Sullivan, F.: Geometrical properties determined by the higher duals of a Banach space. Illinois J. Math. 21, 315–331 (1977)
Taylor, A.E.: The extension of linear functionals. Duke Math. J. 5, 538–547 (1939)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-020-00831-5