On Fréchet differentiability of convex functions on Banach spaces

Wee-Kee Tang

Commentationes Mathematicae Universitatis Carolinae (1995)

  • Volume: 36, Issue: 2, page 249-253
  • ISSN: 0010-2628

Abstract

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Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz function f defined on a separable Banach space are studied. The conditions are in terms of a majorization of f by a C 1 -smooth function, separability of the boundary for f or an approximation of f by Fréchet smooth convex functions.

How to cite

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Tang, Wee-Kee. "On Fréchet differentiability of convex functions on Banach spaces." Commentationes Mathematicae Universitatis Carolinae 36.2 (1995): 249-253. <http://eudml.org/doc/247708>.

@article{Tang1995,
abstract = {Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz function $f$ defined on a separable Banach space are studied. The conditions are in terms of a majorization of $f$ by a $C^1$-smooth function, separability of the boundary for $f$ or an approximation of $f$ by Fréchet smooth convex functions.},
author = {Tang, Wee-Kee},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Fréchet differentiability; convex functions; variational principles; Asplund spaces; separability of the range of the subdifferential; convex Lipschitz function; -smooth function; Fréchet smooth convex functions},
language = {eng},
number = {2},
pages = {249-253},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On Fréchet differentiability of convex functions on Banach spaces},
url = {http://eudml.org/doc/247708},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Tang, Wee-Kee
TI - On Fréchet differentiability of convex functions on Banach spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 2
SP - 249
EP - 253
AB - Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz function $f$ defined on a separable Banach space are studied. The conditions are in terms of a majorization of $f$ by a $C^1$-smooth function, separability of the boundary for $f$ or an approximation of $f$ by Fréchet smooth convex functions.
LA - eng
KW - Fréchet differentiability; convex functions; variational principles; Asplund spaces; separability of the range of the subdifferential; convex Lipschitz function; -smooth function; Fréchet smooth convex functions
UR - http://eudml.org/doc/247708
ER -

References

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  1. Asplund E., Rockafellar R.T., Gradients of convex functions, Trans. Amer. Math. Soc. 139 (1968), 443-467. (1968) MR0240621
  2. Deville R., Godefroy G., Zizler V., Renormings and Smoothness in Banach Spaces, Pitman Monograph and Survey in Pure and Applied Mathematics 64. 
  3. Fabian M., On projectional resolution of identity on the duals of certain Banach spaces, Bull. Austral. Math Soc. 35 (1987), 363-371. (1987) MR0888895
  4. Godefroy G., Some applications of Simons' inequality, Seminar of Functional Analysis II, Univ. of Murcia, to appear. MR1767034
  5. McLaughlin D., Poliquin R.A., Vanderwerff J.D., Zizler V.E., Second order Gateaux differentiable bump functions and approximations in Banach spaces, Can. J. Math 45:3 (1993), 612-625. (1993) Zbl0796.46005MR1222519
  6. Phelps R.R., Convex Functions, Monotone Operators and Differentiability, Lect. Notes in Math., Springer-Verlag 1364 (1993) (Second Edition). Zbl0921.46039MR1238715
  7. Preiss D., Zajíček D., Fréchet differentiation of convex functions in Banach space with separable dual, Proc. Amer. Math. Soc. 91 (1984), 202-204. (1984) MR0740171
  8. Simons S., A convergence theorem with boundary, Pacific J. Math. 40 (1972), 703-708. (1972) Zbl0237.46012MR0312193

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