A remark on totally smooth renormings. (English) Zbl 1445.46008
In [E. Oja et al., Arch. Math. 112, No. 3, 269–281 (2019; Zbl 1415.46010)], a paper coauthored by the reviewer, we proved that every WCG space with a Hahn-Banach smooth norm admits a totally smooth renorming. (\((X,\|\;.\;\|)\) is totally smooth if, for every subspace \(U\subset X\), every functional \(u^*\in U^*\) has a unique norm preserving extension to a functional \(x^{***}\in X^{***}\).) Here, the authors note that such a renorming is possible for every Hahn-Banach smooth space by using a result of M. Raja [Isr. J. Math. 129, 77–91 (2002; Zbl 1024.46004)]; so the WCG assumption is not needed at all.
Reviewer’s remark: M. Cúth and R. Smith have kindly pointed out to me (independently of the paper under review) the same extension of our result, this time relying on M. Raja’s paper [Mathematika 46, No. 2, 343–358 (1999; Zbl 1031.46022)].
Reviewer’s remark: M. Cúth and R. Smith have kindly pointed out to me (independently of the paper under review) the same extension of our result, this time relying on M. Raja’s paper [Mathematika 46, No. 2, 343–358 (1999; Zbl 1031.46022)].
Reviewer: Dirk Werner (Berlin)
MSC:
| 46B03 | Isomorphic theory (including renorming) of Banach spaces |
| 46B04 | Isometric theory of Banach spaces |
| 46B20 | Geometry and structure of normed linear spaces |
| 46B26 | Nonseparable Banach spaces |
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