Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the continuity of biconjugate convex functions

Authors: J. M. Borwein and J. D. Vanderwerff
Journal: Proc. Amer. Math. Soc. 130 (2002), 1797-1803
MSC (2000): Primary 46B20, 52A41
Published electronically: October 24, 2001
MathSciNet review: 1887028
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that a Banach space is a Grothendieck space if and only if every continuous convex function on $X$ has a continuous biconjugate function on $X^{**}$, thus also answering a question raised by S. Simons. Related characterizations and examples are given.

References [Enhancements On Off] (What's this?)

  • 1. H. Attouch and G. Beer, On the convergence of subdifferentials of convex functions, Arch. Math. 60 (1993), 389-400. MR 94b:49018
  • 2. J. Borwein, M. Fabian and J. Vanderwerff, Characterizations of Banach spaces via convex and other locally Lipschitz functions, Acta. Vietnamica 22 (1997), 53-69. MR 98k:46019
  • 3. J. Borwein, S. Fitzpatrick and J. Vanderwerff, Examples of convex functions and classifications of normed spaces, J. Convex Anal. 1 (1994), 61-73. MR 96e:46012
  • 4. R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitnam Monographs and Surveys in Pure and Applied Mathematics 64, Longman, 1993. MR 94d:46012
  • 5. J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics 92, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984. MR 85i:46020
  • 6. J. Diestel and J.J. Uhl, Jr., Vector Measures, AMS Mathematical Surveys 15, Providence, 1977. MR 56:12216
  • 7. M.J. Fabian, Gâteaux Differentiability of Convex Functions and Topology, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., 1997. MR 98h:46009
  • 8. G. Godefroy, Prolongement de fonctions convexes définies sur un espace de Banach $E$ au bidual $E^{\prime \prime }$, C. R. Acad. Sci. Paris 292 (1981), 371-374. MR 82b:46022
  • 9. R. Haydon, A non-reflexive Grothendieck space that does not contain $\ell _{\infty }$, Israel J. Math. 40 (1981), 65-73. MR 83a:46028
  • 10. R.B. Holmes, Geometric Functional Analysis and its Applications, Graduate Texts in Mathematics 24, Springer-Verlag, New York, Heildelberg, Berlin, 1975. MR 53:14085
  • 11. H.H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, New York, Heidelberg, Berlin, 1974. MR 54:11023
  • 12. S. Simons, Maximal monotone multifunctions of Brønsdsted-Rockafellar type, Set-Valued Analysis 7 (1999), 255-294. MR 2001b:47100
  • 13. M. Talagrand, Un nouveau $C(K)$ qui possède la propriété de Grothendieck, Israel J. Math. 37 (1980), 181-191. MR 82g:46029

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46B20, 52A41

Retrieve articles in all journals with MSC (2000): 46B20, 52A41

Additional Information

J. M. Borwein
Affiliation: Department of Mathematics & Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

J. D. Vanderwerff
Affiliation: Department of Mathematics, La Sierra University, Riverside, California 92515

Keywords: Continuous convex function, conjugate function, Grothendieck space
Received by editor(s): September 11, 2000
Received by editor(s) in revised form: January 9, 2001
Published electronically: October 24, 2001
Additional Notes: The first author’s research was supported by an NSERC grant
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society