Every separable infinite-dimensional superreflexive Banach space admits an equivalent norm which is Fréchet differentiable only on an Aronszajn null set.
@article{bwmeta1.element.bwnjournal-article-smv133i1p93bwm, author = {Eva Matou\v skov\'a}, title = {An almost nowhere Fr\'echet smooth norm on superreflexive spaces}, journal = {Studia Mathematica}, volume = {133}, year = {1999}, pages = {93-99}, zbl = {0923.46016}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv133i1p93bwm} }
Matoušková, Eva. An almost nowhere Fréchet smooth norm on superreflexive spaces. Studia Mathematica, Tome 133 (1999) pp. 93-99. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv133i1p93bwm/
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