We show that on every nonseparable Banach space which has a fundamental system (e.g\. on every nonseparable weakly compactly generated space, in particular on every nonseparable Hilbert space) there is a convex continuous function $f$ such that the set of its G\^ateaux differentiability points is not Borel. Thereby we answer a question of J. Rainwater (1990) and extend, in the same time, a former result of M. Talagrand (1979), who gave an example of such a function $f$ on $\ell^1(\frak c)$.
@article{119025,
author = {Petr Holick\'y and M. \v Sm\'\i dek and Lud\v ek Zaj\'\i \v cek},
title = {Convex functions with non-Borel set of G\^ateaux differentiability points},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {39},
year = {1998},
pages = {469-482},
zbl = {0970.46026},
mrnumber = {1666778},
language = {en},
url = {http://dml.mathdoc.fr/item/119025}
}
Holický, Petr; Šmídek, M.; Zajíček, Luděk. Convex functions with non-Borel set of Gâteaux differentiability points. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) pp. 469-482. http://gdmltest.u-ga.fr/item/119025/
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