Residual norm attainment and differentiability of the dual norm
Let $(X, \|\cdot\|)$ be a real Banach space with closed unit ball $B_X$ and dual norm $\|\cdot\|_*$ on $X^*$, and let $$\mathrm{NA}(X) = \{f \in X^* : f(x) = \|f\|_* \text{ for some } x \in B_X\}.$$ By the Bishop–Phelps theorem $\mathrm{NA}(X)$ is norm dense in $X^*$. Guirao, Montesinos and Zizler characterized, for individual norms on separable Banach spaces, when $\mathrm{NA}(X)$ is residual in terms of Fréchet differentiability of the dual norm. Moors and Tan proved that the implication below holds for dual differentiation spaces, a class that includes spaces admitting an equivalent locally uniformly rotund norm, spaces with the Radon–Nikodym property, spaces whose dual is weak Asplund, and spaces that are Lindelöf in the weak topology.
Problem. Let $X$ be an arbitrary real Banach space, and suppose that $\mathrm{NA}(X)$ is residual in the norm topology of $X^*$. Must $\|\cdot\|_*$ be Fréchet differentiable at every point of some norm-dense subset of $X^*$?
Source: Borwein and Moors (2007), Question 2.1 (question 1128 of Open Problems in Topology II).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, Banach spaces, Fréchet differentiability, norm-attaining functionals
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@incollection{borwein2007,
author = {Borwein, Jonathan M. and Moors, Warren B.},
title = {Non-smooth analysis, optimisation theory and {B}anach space theory},
booktitle = {Open Problems in Topology {II}},
editor = {Pearl, Elliott},
publisher = {Elsevier},
address = {Amsterdam},
year = {2007},
pages = {549--559},
doi = {10.1016/B978-044452208-5/50050-8},
}
@article{bishop1961,
author = {Bishop, Errett and Phelps, R. R.},
title = {A proof that every {B}anach space is subreflexive},
journal = {Bull. Amer. Math. Soc.},
volume = {67},
number = {1},
year = {1961},
pages = {97--98},
doi = {10.1090/S0002-9904-1961-10514-4},
}
@article{guirao2018,
author = {Guirao, Antonio J. and Montesinos, Vicente and Zizler, Vaclav},
title = {Remarks on the set of norm-attaining functionals and differentiability},
journal = {Studia Math.},
volume = {241},
number = {1},
year = {2018},
pages = {71--86},
doi = {10.4064/sm8768-6-2017},
}
@article{moors2019,
author = {Moors, Warren B. and Tan, Ne{\c{s}}et {\"O}zkan},
title = {Dual differentiation spaces},
journal = {Bull. Aust. Math. Soc.},
volume = {99},
number = {3},
year = {2019},
pages = {467--472},
doi = {10.1017/S0004972719000376},
}
@misc{aim-residual-norm-attainment-and-differentiability-of-the-dual-norm,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Residual norm attainment and differentiability of the dual norm}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/731-residual-norm-attainment-and-differentiability-of-the-dual-norm.md},
note = {Entry 731 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01122, "Residual norm attainment and differentiability of the dual norm", clopen.solutions.