Convex functions with non-Borel set of Gâteaux differentiability points

Petr Holický; M. Šmídek; Luděk Zajíček

Commentationes Mathematicae Universitatis Carolinae (1998)

  • Volume: 39, Issue: 3, page 469-482
  • ISSN: 0010-2628

Abstract

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We show that on every nonseparable Banach space which has a fundamental system (e.gȯn 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 Gâteaux 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 ℓ 1 ( 𝔠 ) .

How to cite

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Holický, Petr, Šmídek, M., and Zajíček, Luděk. "Convex functions with non-Borel set of Gâteaux differentiability points." Commentationes Mathematicae Universitatis Carolinae 39.3 (1998): 469-482. <http://eudml.org/doc/248238>.

@article{Holický1998,
abstract = {We show that on every nonseparable Banach space which has a fundamental system (e.gȯn 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 Gâteaux 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 $\ell ^1(\mathfrak \{c\})$.},
author = {Holický, Petr, Šmídek, M., Zajíček, Luděk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {convex function; Gâteaux differentiability points; Borel set; fundamental system; convex function; Gâteaux differentiability points; Borel set; fundamental system},
language = {eng},
number = {3},
pages = {469-482},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Convex functions with non-Borel set of Gâteaux differentiability points},
url = {http://eudml.org/doc/248238},
volume = {39},
year = {1998},
}

TY - JOUR
AU - Holický, Petr
AU - Šmídek, M.
AU - Zajíček, Luděk
TI - Convex functions with non-Borel set of Gâteaux differentiability points
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 3
SP - 469
EP - 482
AB - We show that on every nonseparable Banach space which has a fundamental system (e.gȯn 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 Gâteaux 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 $\ell ^1(\mathfrak {c})$.
LA - eng
KW - convex function; Gâteaux differentiability points; Borel set; fundamental system; convex function; Gâteaux differentiability points; Borel set; fundamental system
UR - http://eudml.org/doc/248238
ER -

References

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