Geodesic
Convex functions with non-Borel set of Gâteaux differentiability points
Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 3, pp. 469-482 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\ell^1(\frak c)$.
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 $\ell^1(\frak c)$.
Classification : 46B20, 46B26, 46G05
Keywords: convex function; G\^ateaux differentiability points; Borel set; fundamental system 
@article{CMUC_1998_39_3_a3,
     author = {Holick\'y, P. and \v{S}m{\'\i}dek, M. and Zaj{\'\i}\v{c}ek, L.},
     title = {Convex functions with {non-Borel} set of {G\^ateaux} differentiability points},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {469--482},
     year = {1998},
     volume = {39},
     number = {3},
     mrnumber = {1666778},
     zbl = {0970.46026},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1998_39_3_a3/}
}
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Holický, P.; Šmídek, M.; Zajíček, L. Convex functions with non-Borel set of Gâteaux differentiability points. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 3, pp. 469-482. http://geodesic.mathdoc.fr/item/CMUC_1998_39_3_a3/